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Eigenvalues of a low-rank matrix. Suppose that \( A \in {M}_{n} \) is factored as \( A = X{Y}^{T} \), in which \( X, Y \in {M}_{n, r} \) and \( r < n \) . Then the eigenvalues of \( A \) are the same as those of the \( r \) -by- \( r \) matrix \( {Y}^{T}X \), together with \( n - r \) zeroes. | For example, consider the \( n \) -by- \( n \) all-ones matrix \( {J}_{n} = e{e}^{T} \) (0.2.8). Its eigenvalues are the eigenvalue of the 1-by-1 matrix \( {e}^{T}e = \left\lbrack n\right\rbrack \), namely, \( n \), together with \( n - 1 \) zeroes. The eigenvalues of any matrix of the form \( A = x{y}^{T} \) with \( x... | No |
Cauchy’s determinant identity. Let a nonsingular \( A \in {M}_{n} \) and \( x, y \in {\mathbf{C}}^{n} \) be given. Then\n\n\[ \det \left( {A + x{y}^{T}}\right) = \left( {\det A}\right) \left( {\det \left( {I + {A}^{-1}x{y}^{T}}\right) }\right) \] | \[ \det \left( {A + x{y}^{T}}\right) = \left( {\det A}\right) \mathop{\prod }\limits_{{i = 1}}^{n}{\lambda }_{i}\left( {I + {A}^{-1}x{y}^{T}}\right) \]\n\n\[ = \left( {\det A}\right) \mathop{\prod }\limits_{{i = 1}}^{n}\left( {1 + {\lambda }_{i}\left( {{A}^{-1}x{y}^{T}}\right) }\right) \]\n\n\[ = \left( {\det A}\right)... | Yes |
For any \( n \geq 2 \), consider the \( n \) -by- \( n \) real symmetric Hankel matrix\n\n\[ A = {\left\lbrack i + j\right\rbrack }_{i, j = 1}^{n} = \left\lbrack \begin{matrix} 2 & 3 & 4 & \cdots \\ 3 & 4 & 5 & \cdots \\ 4 & 5 & 6 & \cdots \\ \vdots & & & \ddots \end{matrix}\right\rbrack = v{e}^{T} + e{v}^{T} = \left\l... | According to (1.2.4b), the eigenvalues \( B \) (one positive and one negative) are\n\n\[ n\left( {n + 1}\right) \left\lbrack {\frac{1}{2} \pm \sqrt{\frac{{2n} + 1}{6\left( {n + 1}\right) }}}\right\rbrack \] | Yes |
For any \( n \geq 2 \), consider the \( n \) -by- \( n \) real skew-symmetric Toeplitz matrix\n\n\[ A = {\left\lbrack i - j\right\rbrack }_{i, j = 1}^{n} = \left\lbrack \begin{matrix} 0 & - 1 & - 2 & \cdots \\ 1 & 0 & - 1 & \cdots \\ 2 & 1 & 0 & \cdots \\ \vdots & & & \ddots \end{matrix}\right\rbrack = v{e}^{T} - e{v}^... | which, using (1.2.4b) again, are \( \pm \frac{ni}{2}\sqrt{\frac{{n}^{2} - 1}{3}} \) . | Yes |
Theorem 1.3.27. Suppose that \( A \in {M}_{n} \) is diagonalizable, let \( {\mu }_{1},\ldots ,{\mu }_{d} \) be its distinct eigenvalues with respective multiplicities \( {n}_{1},\ldots ,{n}_{d} \), let \( S, T \in {M}_{n} \) be nonsingular, and suppose that \( A = {S\Lambda }{S}^{-1} \), in which \( \Lambda \) is a dia... | Proof. We have \( {S\Lambda }{S}^{-1} = {T\Lambda }{T}^{-1} \) if and only if \( \left( {{S}^{-1}T}\right) \Lambda = \Lambda \left( {{S}^{-1}T}\right) \) if and only if \( {S}^{-1}T \) is block diagonal conformal to \( \Lambda \) (0.7.7), that is, if and only if \( {S}^{-1}T = \) \( {R}_{1} \oplus \cdots \oplus {R}_{d}... | Yes |
Lemma 1.3.28. Let \( S \in {M}_{n} \) be nonsingular and let \( S = C + {iD} \), in which \( C, D \in \) \( {M}_{n}\left( \mathbf{R}\right) \) . There is a real number \( \tau \) such that \( T = C + {\tau D} \) is nonsingular. | Proof. If \( C \) is nonsingular, take \( \alpha = 0 \) . If \( C \) is singular, consider the polynomial \( p\left( t\right) = \det \left( {C + {tD}}\right) \), which is not a constant (degree zero) polynomial since \( p\left( 0\right) = \) \( \det C = 0 \neq \det S = p\left( i\right) \) . Since \( p\left( t\right) \)... | Yes |
Theorem 1.3.29. Let \( \mathcal{F} = \left\{ {{A}_{\alpha } : \alpha \in \mathcal{I}}\right\} \subset {M}_{n}\left( \mathbf{R}\right) \) and \( \mathcal{G} = \left\{ {{B}_{\alpha } : \alpha \in \mathcal{I}}\right\} \subset {M}_{n}\left( \mathbf{R}\right) \) be given families of real matrices. If there is a nonsingular ... | Proof. Let \( S = C + {iD} \) be nonsingular, in which \( C, D \in {M}_{n}\left( \mathbf{R}\right) \) . The preceding lemma ensures that there is a real number \( \tau \) such that \( T = C + {\tau D} \) is nonsingular. The similarity \( {A}_{\alpha } = S{B}_{\alpha }{S}^{-1} \) is equivalent to the identity \( {A}_{\a... | Yes |
Corollary 1.3.30. Let \( \mathcal{F} = \left\{ {{A}_{\alpha } : \alpha \in \mathcal{I}}\right\} \subset {M}_{n}\left( \mathbf{R}\right) \) be a family of real diagonalizable matrices with real eigenvalues. Then \( \mathcal{F} \) is a commuting family if and only if there is a nonsingular real matrix \( T \) such that \... | Proof. For the \ | No |
Theorem 1.4.9. Let \( A, B \in {M}_{n} \) and suppose that \( B = {S}^{-1}{AS} \) for some nonsingular \( S \) . If \( x \in {\mathbf{C}}^{n} \) is a right eigenvector of \( B \) associated with an eigenvalue \( \lambda \), then \( {Sx} \) is a right eigenvector of \( A \) associated with \( \lambda \) . If \( y \in {\... | Proof. If \( {Bx} = {\lambda x} \), then \( {S}^{-1}{ASx} = {\lambda x} \), or \( A\left( {Sx}\right) = \lambda \left( {Sx}\right) \) . Since \( S \) is nonsingular and \( x \neq 0,{Sx} \neq 0 \), and hence \( {Sx} \) is an eigenvector of \( A \) . If \( {y}^{ * }B = \lambda {y}^{ * } \), then \( {y}^{ * }{S}^{-1}{AS} ... | Yes |
Theorem 1.4.10. Let \( A \in {M}_{n} \) and \( \lambda \in \mathbf{C} \) be given, and let \( k \geq 1 \) be a given positive integer. Consider the following three statements:\n\n(a) \( \lambda \) is an eigenvalue of \( A \) with geometric multiplicity at least \( k \) .\n\n(b) For each \( m = n - k + 1,\ldots, n,\lamb... | Proof. (a) \( \Rightarrow \) (b): Let \( \lambda \) be an eigenvalue of \( A \) with geometric multiplicity at least \( k \) , which means that \( \operatorname{rank}\left( {A - {\lambda I}}\right) \leq n - k \) . Suppose that \( m > n - k \) . Then every \( m \) -by- \( m \) minor of \( A - {\lambda I} \) is zero. In ... | Yes |
Lemma 1.4.11. Let \( A \in {M}_{n},\lambda \in \mathbf{C} \), and nonzero vectors \( x, y \in {\mathbf{C}}^{n} \) be given. Suppose that \( \lambda \) has geometric multiplicity \( I \) as an eigenvalue of \( A,{Ax} = {\lambda x} \), and \( {y}^{ * }A = \lambda {y}^{ * } \) . Then there is a nonzero \( \gamma \in \math... | Proof. We have \( \operatorname{rank}\left( {{\lambda I} - A}\right) = n - 1 \) and hence \( \operatorname{rank}\operatorname{adj}\left( {{\lambda I} - A}\right) = 1 \), that is, \( \operatorname{adj}\left( {{\lambda I} - A}\right) = \xi {\eta }^{ * } \) for some nonzero \( \xi ,\eta \in {\mathbf{C}}^{n} \) ; see (0.8.... | Yes |
Theorem 1.4.12. Let \( A \in {M}_{n},\lambda \in \mathbf{C} \), and nonzero vectors \( x, y \in {\mathbf{C}}^{n} \) be given. Suppose that \( \lambda \) is an eigenvalue of \( A,{Ax} = {\lambda x} \), and \( {y}^{ * }A = \lambda {y}^{ * } \) . (a) If \( \lambda \) has algebraic multiplicity 1, then \( {y}^{ * }x \neq 0... | Proof. In both cases (a) and (b), \( \lambda \) has geometric multiplicity 1 ; the preceding lemma tells us that there is a nonzero \( \gamma \in \mathbf{C} \) such that \( \operatorname{adj}\left( {{\lambda I} - A}\right) = {\gamma x}{y}^{ * } \) . Then \( {p}_{A}\left( \lambda \right) = 0 \) and \( {p}_{A}^{\prime }\... | Yes |
Theorem 2.1.2. Every orthonormal list of vectors in \( {\mathbf{C}}^{n} \) is linearly independent. | Proof. Suppose that \( \left\{ {{x}_{1},\ldots ,{x}_{k}}\right\} \) is an orthonormal set, and suppose that \( 0 = {\alpha }_{1}{x}_{1} + \) \( \cdots + {\alpha }_{k}{x}_{k}. \) Then \( 0 = {\left( {\alpha }_{1}{x}_{1} + \cdots + {\alpha }_{k}{x}_{k}\right) }^{ * }\left( {{\alpha }_{1}{x}_{1} + \cdots + {\alpha }_{k}{x... | Yes |
Theorem 2.1.4. If \( U \in {M}_{n} \), the following are equivalent:\n\n(a) \( U \) is unitary.\n\n(b) \( U \) is nonsingular and \( {U}^{ * } = {U}^{-1} \).\n\n(c) \( U{U}^{ * } = I \).\n\n(d) \( {U}^{ * } \) is unitary.\n\n(e) The columns of \( U \) are orthonormal.\n\n(f) The rows of \( U \) are orthonormal.\n\n(g) ... | Proof. (a) implies (b) since \( {U}^{-1} \) (when it exists) is the unique matrix, left multiplication by which produces \( I\left( {0.5}\right) \) ; the definition of unitary says that \( {U}^{ * } \) is such a matrix. Since \( {BA} = I \) if and only if \( {AB} = I \) (for \( A, B \in {M}_{n}\left( {0.5}\right) \) ),... | Yes |
Lemma 2.1.8. Let \( {U}_{1},{U}_{2},\ldots \in {M}_{n} \) be a given infinite sequence of unitary matrices. There exists an infinite subsequence \( {U}_{{k}_{1}},{U}_{{k}_{2}},\ldots ,1 \leq {k}_{1} < {k}_{2} < \cdots \), such that all of the entries of \( {U}_{{k}_{i}} \) converge (as sequences of complex numbers) to ... | Proof. All that is required here is the fact that from any infinite sequence in a compact set, one may always select a convergent subsequence. We have already observed that if a sequence of unitary matrices converges to some matrix, then the limit matrix must be unitary. | No |
Theorem 2.1.9. Let \( A \in {M}_{n} \) be nonsingular. Then \( {A}^{-1} \) is similar to \( {A}^{ * } \) if and only if there is a nonsingular \( B \in {M}_{n} \) such that \( A = {B}^{-1}{B}^{ * } \) . | Proof. If \( A = {B}^{-1}{B}^{ * } \) for some nonsingular \( B \in {M}_{n} \), then \( {A}^{-1} = {\left( {B}^{ * }\right) }^{-1}B \) and \( {B}^{ * }{A}^{-1}{\left( {B}^{ * }\right) }^{-1} = B{\left( {B}^{ * }\right) }^{-1} = {\left( {B}^{-1}{B}^{ * }\right) }^{ * } = {A}^{ * } \), so \( {A}^{-1} \) is similar to \( ... | Yes |
Theorem 2.1.13. Let \( x, y \in {\mathbf{C}}^{n} \) be given and suppose that \( \parallel x{\parallel }_{2} = \parallel y{\parallel }_{2} > 0 \) . If \( y = \) \( {e}^{i\theta }x \) for some real \( \theta \), let \( U\left( {y, x}\right) = {e}^{i\theta }{I}_{n} \) ; otherwise, let \( \phi \in \lbrack 0,{2\pi }) \) be... | Proof. The assertions are readily verified if \( x \) and \( y \) are linearly dependent, that is, if \( y = {e}^{i\theta }x \) for some real \( \theta \) . If \( x \) and \( y \) are linearly independent, the Cauchy-Schwarz inequality (0.6.3) ensures that \( {x}^{ * }x \neq \left| {{x}^{ * }y}\right| \) . Compute\n\n\... | Yes |
Theorem 2.1.14 ( \( {QR} \) factorization). Let \( A \in {M}_{n, m} \) be given.\n\n(a) If \( n \geq m \), there is a \( Q \in {M}_{n, m} \) with orthonormal columns and an upper triangular \( R \in {M}_{m} \) with nonnegative main diagonal entries such that \( A = {QR} \) . | Proof. Let \( {a}_{1} \in {\mathbf{C}}^{n} \) be the first column of \( A \), let \( {r}_{1} = {\begin{Vmatrix}{a}_{1}\end{Vmatrix}}_{2} \), and let \( {U}_{1} \) be a unitary matrix such that \( {U}_{1}{a}_{1} = {r}_{1}{e}_{1} \) . Theorem 2.1.13 gives an explicit construction for such a matrix, which is either a unit... | Yes |
Theorem 2.1.18. If \( X = \left\lbrack \begin{array}{lll} {x}_{1} & \ldots & {x}_{k} \end{array}\right\rbrack \in {M}_{n, k} \) and \( Y = \left\lbrack \begin{array}{lll} {y}_{1} & \ldots & {y}_{k} \end{array}\right\rbrack \in {M}_{n, k} \) have orthonormal columns, then there is a unitary \( U \in {M}_{n} \) such that... | Proof. Extend each of the orthonormal lists \( {x}_{1},\ldots ,{x}_{k} \) and \( {y}_{1},\ldots ,{y}_{k} \) to orthonormal bases of \( {\mathbf{C}}^{n} \) ; see (0.6.4-5). That is, construct unitary matrices \( V = \left\lbrack \begin{array}{ll} X & {X}_{2} \end{array}\right\rbrack \) and \( W = \left\lbrack \begin{arr... | Yes |
Theorem 2.2.2. Let \( U \in {M}_{n} \) and \( V \in {M}_{m} \) be unitary, let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n, m} \) and \( B = \left\lbrack {b}_{ij}\right\rbrack \in {M}_{n, m} \), and suppose that \( A = {UBV} \). Then \( \mathop{\sum }\limits_{{i, j = 1}}^{{n, m}}{\left| {b}_{ij}\right| }^{2} =... | Proof. It suffices to check that \( \operatorname{tr}{B}^{ * }B = \operatorname{tr}{A}^{ * }A \) ; see (0.2.5). Compute \( \operatorname{tr}{A}^{ * }A = \) \( \operatorname{tr}{\left( UBV\right) }^{ * }\left( {UBV}\right) = \operatorname{tr}\left( {{V}^{ * }{B}^{ * }{U}^{ * }{UBV}}\right) = \operatorname{tr}{V}^{ * }{B... | Yes |
Unitary similarity to a matrix with equal diagonal entries. Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) be given. We claim that there is a unitary \( U \in {M}_{n} \) such that all the main diagonal entries of \( {U}^{ * }{AU} = B = \left\lbrack {b}_{ij}\right\rbrack \) are equal; if \( A \) is real, t... | Begin by considering the complex case and \( n = 2 \) . Since we can replace \( A \in {M}_{2} \) by \( A - \left( {\frac{1}{2}\operatorname{tr}A}\right) I \), there is no loss of generality to assume that \( \operatorname{tr}A = 0 \) , in which case the two eigenvalues of \( A \) are \( \pm \lambda \) for some \( \lamb... | Yes |
Example 2.2.4. Unitary similarity to an upper Hessenberg matrix. Let \( A = \) \( \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) be given. The following construction shows that \( A \) is unitarily similar to an upper Hessenberg matrix with nonnegative entries in its first subdiagonal. | Let \( {a}_{1} \) be the first column of \( A \), partitioned as \( {a}_{1}^{T} = \left\lbrack {{a}_{11}{\xi }^{T}}\right\rbrack \) with \( \xi \in {\mathbf{C}}^{n - 1} \) . Let \( {U}_{1} = {I}_{n - 1} \) if \( \xi = 0 \) ; otherwise, use (2.1.13) to construct \( {U}_{1} = U\left( {\parallel \xi {\parallel }_{2}{e}_{1... | Yes |
Theorem 2.3.1 (Schur form; Schur triangularization). Let \( A \in {M}_{n} \) have eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) in any prescribed order and let \( x \in {\mathbf{C}}^{n} \) be a unit vector such that \( {Ax} = {\lambda }_{1}x \n\n(a) There is a unitary \( U = \left\lbrack \begin{array}{llll} x... | Proof. Let \( x \) be a normalized eigenvector of \( A \) associated with the eigenvalue \( {\lambda }_{1} \), that is, \( {x}^{ * }x = 1 \) and \( {Ax} = {\lambda }_{1}x \) . Let \( {U}_{1} = \left\lbrack \begin{array}{llll} x & {u}_{2} & \ldots & {u}_{n} \end{array}\right\rbrack \) be any unitary matrix whose first c... | Yes |
Example 2.3.2. If the eigenvalues of \( A \) are reordered and the corresponding upper triangularization (2.3.1) is performed, the entries of \( T \) above the main diagonal can be different. Consider\n\n\[ \n{T}_{1} = \left\lbrack \begin{array}{lll} 1 & 1 & 4 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{array}\right\rbrack ,{T}_{2}... | \[ \n{T}_{1} = \left\lbrack \begin{array}{lll} 1 & 1 & 4 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{array}\right\rbrack ,{T}_{2} = \left\lbrack \begin{matrix} 2 & - 1 & 3\sqrt{2} \\ 0 & 1 & \sqrt{2} \\ 0 & 0 & 3 \end{matrix}\right\rbrack, U = \frac{1}{\sqrt{2}}\left\lbrack \begin{matrix} 1 & 1 & 0 \\ 1 & - 1 & 0 \\ 0 & 0 & \sqrt{2... | Yes |
Theorem 2.3.3. Let \( \mathcal{F} \subseteq {M}_{n} \) be a nonempty commuting family. There is a unitary \( U \in {M}_{n} \) such that \( {U}^{ * }{AU} \) is upper triangular for every \( A \in \mathcal{F} \) . | Proof. Return to the proof of (2.3.1). Exploiting (1.3.19) at each step of the proof in which a choice of an eigenvector (and unitary matrix) is made, choose a unit eigenvector that is common to every \( A \in \mathcal{F} \) and construct a unitary matrix that has this common eigenvector as its first column; it deflate... | Yes |
Theorem 2.3.4 (real Schur form). Let \( A \in {M}_{n}\left( \mathbf{R}\right) \) be given.\n\n(a) There is a real nonsingular \( S \in {M}_{n}\left( \mathbf{R}\right) \) such that \( {S}^{-1}{AS} \) is a real upper quasi-triangular matrix\n\n\[ \left\lbrack \begin{matrix} {A}_{1} & & & \star \\ & {A}_{2} & & \\ & & \dd... | Proof. (a) The proof of (2.3.1) shows how to deflate \( A \) by a real orthogonal similarity corresponding to any given real eigenpair; that deflation produces a real 1-by-1 diagonal block and a deflated matrix of the form \( \left\lbrack \begin{array}{ll} \lambda & * \\ 0 & \mathcal{A} \end{array}\right\rbrack \) . Pr... | Yes |
Theorem 2.3.6. Let \( \mathcal{F} \subseteq {M}_{n}\left( \mathbf{R}\right) \) be a nonempty commuting family.\n\n(a) There is a nonsingular \( S \in {M}_{n}\left( \mathbf{R}\right) \) and a quasidiagonal \( D = {J}_{{n}_{1}} \oplus \cdots \oplus {J}_{{n}_{m}} \in \) \( {M}_{n} \) such that: (i) for each \( A \in \math... | Proof. (a) Following the inductive pattern of the proof of (2.3.3), it suffices to construct a nonsingular real matrix that deflates (via similarity) each matrix in \( \mathcal{F} \) in the same way. Use (1.3.19) to choose a common unit eigenvector \( x \in {\mathbf{C}}^{n} \) of every \( A \in \mathcal{F} \) . Write \... | Yes |
Corollary 2.3.7. Let \( A \in {M}_{n} \) and suppose that \( A\bar{A} = \bar{A}A \) . There is a real orthogonal \( Q \in {M}_{n}\left( \mathbf{R}\right) \) and a quasidiagonal \( D = {J}_{{n}_{1}} \oplus \cdots \oplus {J}_{{n}_{m}} \in {M}_{n} \) such that \( {Q}^{T}{AQ} \in {M}_{n} \) is a complex upper quasitriangul... | Proof. Write \( A = B + {iC} \), in which \( B \) and \( C \) are real. The hypothesis and the preceding exercise ensure that \( B \) and \( C \) commute. It follows from (2.3.6b) that there is a real orthogonal \( Q \in {M}_{n}\left( \mathbf{R}\right) \) and a quasidiagonal \( D = {J}_{{n}_{1}} \oplus \cdots \oplus {J... | Yes |
Lemma 2.5.2. Let \( A \in {M}_{n} \) be partitioned as \( A = \left\lbrack \begin{matrix} {A}_{11} & {A}_{12} \\ 0 & {A}_{22} \end{matrix}\right\rbrack \), in which \( {A}_{11} \) and \( {A}_{22} \) are square. Then \( A \) is normal if and only if \( {A}_{11} \) and \( {A}_{22} \) are normal and \( {A}_{12} = 0 \) . \... | Proof. If \( {A}_{11} \) and \( {A}_{22} \) are normal and \( {A}_{12} = 0 \), then \( A = {A}_{11} \oplus {A}_{22} \) is a direct sum of normal matrices, so it is normal.\n\nConversely, if \( A \) is normal, then\n\n\[ A{A}^{ * } = \left\lbrack \begin{matrix} {A}_{11}{A}_{11}^{ * } + {A}_{12}{A}_{12}^{ * } & \star \\ ... | Yes |
Theorem 2.5.3. Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) have eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) . The following statements are equivalent:\n\n(a) \( A \) is normal.\n\n(b) \( A \) is unitarily diagonalizable.\n\n(c) \( \mathop{\sum }\limits_{{i, j = 1}}^{n}{\left| {a}_{ij}\right... | Proof. Use (2.3.1) to write \( A = {UT}{U}^{ * } \), in which \( U = \left\lbrack \begin{array}{lll} {u}_{1} & \ldots & {u}_{n} \end{array}\right\rbrack \) is unitary and \( T = \left\lbrack {t}_{ij}\right\rbrack \in {M}_{n} \) is upper triangular.\n\nIf \( A \) is normal, then so is \( T \) (as is every matrix that is... | Yes |
Theorem 2.5.4. Let \( A \in {M}_{n} \) be normal and have distinct eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{d} \), with respective multiplicities \( {n}_{1},\ldots ,{n}_{d} \) . Let \( \Lambda = {\lambda }_{1}{I}_{{n}_{1}} \oplus \cdots \oplus {\lambda }_{d}{I}_{{n}_{d}} \), and suppose that \( U \in {M}_{n} \... | Proof. (a) If \( {U\Lambda }{U}^{ * } = {V\Lambda }{V}^{ * } \), then \( \Lambda {U}^{ * }V = {U}^{ * }{V\Lambda } \), so \( W = {U}^{ * }V \) is unitary and commutes with \( \Lambda \) ; (2.4.4.2) ensures that \( W \) is block diagonal conformal to \( \Lambda \) . Conversely, if \( U = {VW} \) and \( W = {W}_{1} \oplu... | Yes |
Theorem 2.5.5. Let \( \mathcal{N} \subseteq {M}_{n} \) be a nonempty family of normal matrices. Then \( \mathcal{N} \) is a commuting family if and only if it is a simultaneously unitarily diagonalizable family. For any given \( {A}_{0} \in \mathcal{N} \) and for any given ordering \( {\lambda }_{1},\ldots ,{\lambda }_... | Exercise. Use (2.3.3) and the fact that a triangular normal matrix must be diagonal to prove (2.5.5). The final assertion about \( {A}_{0} \) follows as in the proof of (1.3.21) since every permutation matrix is unitary. | No |
Theorem 2.5.6. Let \( A \in {M}_{n} \) be Hermitian and have eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) . Let \( \Lambda = \) \( \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \) . Then\n\n(a) \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) are real.\n\n(b) \( A \) is unitarily diagon... | Proof. A diagonal Hermitian matrix must have real diagonal entries, so (a) follows from (b) and the fact that the set of Hermitian matrices is closed under unitary similarity. Statement (b) follows from (2.5.3) because Hermitian matrices are normal. Statement (c) restates (b) and incorporates the information that the d... | No |
Lemma 2.5.7. Suppose that \( A = \left\lbrack \begin{array}{ll} a & b \\ c & d \end{array}\right\rbrack \in {M}_{2}\left( \mathbf{R}\right) \) is normal and has a conjugate pair of non-real eigenvalues. Then \( c = - b \neq 0 \) and \( d = a \) . | Proof. A computation reveals that \( A{A}^{T} = {A}^{T}A \) if and only if \( {b}^{2} = {c}^{2} \) and \( {ac} + {bd} = \) \( {ab} + {cd} \) . If \( b = c \), then \( A \) is Hermitian (because it is real symmetric), so the preceding theorem ensures that it has two real eigenvalues. Therefore, we must have \( b = - c \... | Yes |
Theorem 2.5.8. Let \( A \in {M}_{n}\left( \mathbf{R}\right) \) be normal.\n\n(a) There is a real orthogonal \( Q \in {M}_{n}\left( \mathbf{R}\right) \) such that \( {Q}^{T}{AQ} \) is a real quasidiagonal matrix\n\n\[ \n{A}_{1} \oplus \cdots \oplus {A}_{m} \in {M}_{n}\left( \mathbf{R}\right) \text{, each}{A}_{i}\text{is... | Proof. (a) Theorem 2.3.4b ensures that \( A \) is real orthogonally similar to a real upper quasitriangular matrix, each of whose 2-by-2 diagonal blocks has a conjugate pair of non-real eigenvalues. Since this upper quasitriangular matrix is normal, (2.5.2) ensures that it is actually quasidiagonal, and each of its 2-b... | Yes |
Corollary 2.5.11. Let \( A \in {M}_{n}\left( \mathbf{R}\right) \) . Then\n\n(a) \( A = {A}^{T} \) if and only if there is a real orthogonal \( Q \in {M}_{n}\left( \mathbf{R}\right) \) such that\n\n\[ \n{Q}^{T}{AQ} = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \in {M}_{n}\left( \mathbf{R}\ri... | Proof. Each of the hypotheses ensures that \( A \) is real and normal, so it is real orthogonally similar to a quasidiagonal matrix of the form (2.5.9). It suffices to consider what each of the hypotheses implies about the direct summands in (2.5.9). If \( A = {A}^{T} \) , there can be no direct summands of the form (2... | Yes |
Theorem 2.5.15. Let \( \mathcal{N} \subseteq {M}_{n}\left( \mathbf{R}\right) \) be a nonempty commuting family of real normal matrices. There is a real orthogonal matrix \( Q \) and a nonnegative integer \( q \) such that, for each \( A \in \mathcal{N},{Q}^{T}{AQ} \) is a real quasidiagonal matrix of the form\n\n\[ \La... | Proof. Theorem 2.3.6b ensures that there is a real orthogonal \( Q \) and a quasidiagonal matrix \( D = {J}_{{n}_{1}} \oplus \cdots \oplus {J}_{{n}_{m}} \) such that for every \( A \in \mathcal{N},{Q}^{T}{AQ} \) is an upper quasitri-angular matrix of the form (2.3.6.1) that is partitioned conformally to \( D \) . Moreo... | Yes |
Theorem 2.5.16 (Fuglede-Putnam). Let \( A \in {M}_{n} \) and \( B \in {M}_{m} \) be normal and let \( X \in {M}_{n, m} \) be given. Then \( {AX} = {XB} \) if and only if \( {A}^{ * }X = X{B}^{ * } \) . | Proof. Let \( A = {U\Lambda }{U}^{ * } \) and \( B = {VM}{V}^{ * } \) be spectral decompositions in which \( \Lambda = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \) and \( M = \operatorname{diag}\left( {{\mu }_{1},\ldots ,{\mu }_{m}}\right) \) . Let \( {U}^{ * }{XV} = \left\lbrack {\xi }_{... | Yes |
Theorem 2.5.17. Let \( A \in {M}_{n} \) be normal. The following three statements are equivalent:\n\n(a) \( \bar{A}A = A\bar{A} \) .\n\n(b) \( {A}^{T}A = A{A}^{T} \) .\n\n(c) There is a real orthogonal \( Q \) such that \( {Q}^{T}{AQ} \) is a direct sum of blocks, in any prescribed order, each of which is either a zero... | Proof. Equivalence of (a) and (b) follows from the preceding theorem: \( \bar{A}A = A\bar{A} \) if and only if \( {A}^{T}A = {\left( \bar{A}\right) }^{ * }A = A{\left( \bar{A}\right) }^{ * } = A{A}^{T} \) .\n\nLet \( A = B + {iC} \), in which \( B \) and \( C \) are real. The exercise following (2.5.16) shows that \( \... | No |
Corollary 2.5.18. Let \( U \in {M}_{n} \) be unitary.\n\n(a) If \( U \) is symmetric, then there is a real orthogonal \( Q \in {M}_{n}\left( \mathbf{R}\right) \) and real \( {\theta }_{1},\ldots ,{\theta }_{n} \in \lbrack 0,{2\pi }) \) such that\n\n\[ U = Q\operatorname{diag}\left( {{e}^{i{\theta }_{1}},\ldots ,{e}^{i{... | Proof. A unitary matrix \( U \) that is either symmetric or skew symmetric satisfies the identity \( U{U}^{T} = {U}^{T}U \), so (2.5.17) ensures that there is a real orthogonal \( Q \) such that \( {Q}^{T}{AQ} \) is a direct sum of nonzero scalar multiples of blocks selected from the four types in (2.5.17.1).\n\n(a) If... | Yes |
Corollary 2.5.20. Let \( U \in {M}_{n} \) be unitary.\n\n(a) If \( U \) is symmetric, there is a unitary symmetric \( V \) such that \( {V}^{2} = U \) and \( V \) is a polynomial in \( U \) . Consequently, \( V \) commutes with any matrix that commutes with \( U \) . | Proof. (a) Use the preceding corollary to factor \( U = P\operatorname{diag}\left( {{e}^{i{\theta }_{1}},\ldots ,{e}^{i{\theta }_{n}}}\right) {P}^{T} \), in which \( P \) is real orthogonal and \( {\theta }_{1},\ldots ,{\theta }_{n} \in \lbrack 0,{2\pi }) \) are real. Let \( p\left( t\right) \) be a polynomial such tha... | Yes |
Theorem 2.5.21. Let \( \mathcal{F} = \left\{ {{A}_{\alpha } : \alpha \in \mathcal{I}}\right\} \subset {M}_{n}\left( \mathbf{R}\right) \) and \( \mathcal{G} = \left\{ {{B}_{\alpha } : \alpha \in \mathcal{I}}\right\} \subset {M}_{n}\left( \mathbf{R}\right) \) be given families of real matrices. If there is a unitary \( U... | Proof. Since each \( {A}_{\alpha } \) and \( {B}_{\alpha } \) is real, \( {A}_{\alpha } = U{B}_{\alpha }{U}^{ * } = \bar{U}{B}_{\alpha }{U}^{T} = {\bar{A}}_{\alpha } \) and hence \( {U}^{T}U{B}_{\alpha } = {B}_{\alpha }{U}^{T}U \) for every \( \alpha \in \mathcal{I} \) . The preceding corollary ensures that there is a ... | Yes |
Theorem 2.6.1. Let \( A, B \in {M}_{n} \) . There are unitary \( V, W \in {M}_{n} \) such that \( A = V{T}_{A}{W}^{ * } \) , \( B = V{T}_{B}{W}^{ * } \), and both \( {T}_{A} \) and \( {T}_{B} \) are upper triangular. If \( B \) is nonsingular, the main diagonal entries of \( {T}_{B}^{-1}{T}_{A} \) are the eigenvalues o... | Proof. Suppose that \( B \) is nonsingular, and use (2.3.1) to write \( {B}^{-1}A = {UT}{U}^{ * } \), in which \( U \) is unitary and \( T \) is upper triangular. Use the \( {QR} \) factorization (2.1.14) to write \( {BU} = {QR} \), in which \( Q \) is unitary and \( R \) is upper triangular. Then \( A = {BUT}{U}^{ * }... | Yes |
Theorem 2.6.2. Let \( A, B \in {M}_{n}\left( \mathbf{R}\right) \) . There are real orthogonal \( V, W \in {M}_{n} \) such that \( A = V{T}_{A}{W}^{T}, B = V{T}_{B}{W}^{T},{T}_{A} \) is real and upper quasitriangular, and \( {T}_{B} \) is real and upper triangular. | Proof. If \( B \) is nonsingular, one uses (2.3.4) to write \( {B}^{-1}A = {UT}{U}^{T} \), in which \( U \) is real orthogonal and \( T \) is real and upper quasitriangular. Use (2.1.14(d)) to write \( {BU} = {QR} \), in which \( Q \) is real orthogonal and \( R \) is real and upper triangular. Then \( {RU} \) is upper... | Yes |
Theorem 2.6.4. Let an infinite sequence \( {A}_{1},{A}_{2},\ldots \in {M}_{n, m} \) be given, suppose that \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{A}_{k} = A \) (entrywise convergence), and let \( q = \min \{ m, n\} \) . Let \( {\sigma }_{1}\left( A\right) \geq \cdots \geq \) \( {\sigma }_{q}\left( A\right) ... | Proof. If the assertion of the theorem is false, then there is some \( {\varepsilon }_{0} > 0 \) and an infinite sequence of positive integers \( {k}_{1} < {k}_{2} < \cdots \) such that for every \( j = 1,2,\ldots \) we have\n\n\[ \mathop{\max }\limits_{{i = 1,\ldots, q}}\left| {{\sigma }_{i}\left( {A}_{{k}_{j}}\right)... | Yes |
Theorem 2.6.5 (Autonne’s uniqueness theorem). Let \( A \in {M}_{n, m} \) be given with \( \operatorname{rank}A = r \) . Let \( {s}_{1},\ldots ,{s}_{d} \) be the distinct positive singular values of \( A \), in any order, with respective multiplicities \( {n}_{1},\ldots ,{n}_{d} \), and let \( {\sum }_{d} = {s}_{1}{I}_{... | Proof. The Hermitian matrix \( {A}^{ * }A \) is represented as \( {A}^{ * }A = {\left( V\sum {W}^{ * }\right) }^{ * }\left( {{V\sum }{W}^{ * }}\right) = \) \( W{\sum }^{T}\sum {W}^{ * } \) and also as \( {A}^{ * }A = \widehat{W}{\sum }^{T}\sum {\widehat{W}}^{ * } \) . Theorem 2.5.4 ensures that there are unitary matric... | Yes |
Corollary 2.6.6. Let \( A \in {M}_{n} \) and let \( r = \operatorname{rank}A \) . (a) (Autonne) \( A = {A}^{T} \) if and only if there is a unitary \( U \in {M}_{n} \) and a nonnegative diagonal matrix \( \sum \) such that \( A = {U\sum }{U}^{T} \) . The diagonal entries of \( \sum \) are the singular values of \( A \)... | Proof. (a) If \( A = {U\sum }{U}^{T} \) for a unitary \( U \in {M}_{n} \) and a nonnegative diagonal matrix \( \sum \) , then \( A \) is symmetric and the diagonal entries of \( \sum \) are its singular values. Conversely, let \( {s}_{1},\ldots ,{s}_{d} \) be the distinct positive singular values of \( A \), in any ord... | Yes |
Corollary 2.6.7. Let \( A \in {M}_{n, m}\left( \mathbf{R}\right) \) and suppose that \( \operatorname{rank}A = r \) . Then \( A = {P\sum }{Q}^{T} \) , in which \( P \in {M}_{n}\left( \mathbf{R}\right) \) and \( Q \in {M}_{m}\left( \mathbf{R}\right) \) are real orthogonal, and \( \sum \in {M}_{n, m}\left( \mathbf{R}\rig... | Proof. Using the notation of (2.6.4), let \( A = {V\sum }{W}^{ * } \) be a given singular value decomposition; the unitary matrices \( V \) and \( W \) need not be real. We have \( {V\sum }{W}^{ * } = \) \( A = \bar{A} = \bar{V}\sum {\bar{W}}^{ * } \), so \( {V}^{T}{V\sum } = \sum {W}^{T}W \) . Theorem 2.6.5 ensures th... | Yes |
Theorem 2.7.1 (CS decomposition). Let \( p, q \), and \( n \) be given integers with \( 1 < p \leq \) \( q < n \) and \( p + q = n \) . Let \( U = \left\lceil \begin{array}{ll} {U}_{11} & {U}_{12} \\ {U}_{21} & {U}_{22} \end{array}\right\rceil \in {M}_{n} \) be unitary, with \( {U}_{11} \in {M}_{p} \) and \( {U}_{22} \... | Proof. Our strategy is to perform a sequence of partitioned unitary equivalences that, step by step, reduce \( U \) to a block matrix that has the asserted form. The first step is to use the singular value decomposition: Write \( {U}_{11} = {V\sum W} = \) \( \left( {V{K}_{p}}\right) \left( {{K}_{p}\sum {K}_{p}}\right) ... | Yes |
Corollary 3.1.21. Let \( A \in {M}_{n} \) and a nonzero \( \epsilon \in \mathbf{C} \) be given. Then there exists a nonsingular \( S\left( \epsilon \right) \in {M}_{n} \) such that\n\n\[ A = S\left( \epsilon \right) \left\lbrack \begin{array}{lll} {J}_{{n}_{1}}\left( {{\lambda }_{1},\epsilon }\right) & & \\ & \ddots & ... | Proof. First find a nonsingular matrix \( {S}_{1} \in {M}_{n} \) such that \( {S}_{1}^{-1}A{S}_{1} \) is a Jordan matrix of the form (3.1.3) (with a real \( {S}_{1} \) if \( A \) is real and has real eigenvalues). Let \( {D}_{\epsilon, i} = \operatorname{diag}\left( {1,\epsilon ,{\epsilon }^{2},\ldots ,{\epsilon }^{{n}... | Yes |
Theorem 3.3.1. Let \( A \in {M}_{n} \) be given. There exists a unique monic polynomial \( {q}_{A}\left( t\right) \) of minimum degree that annihilates \( A \) . The degree of \( {q}_{A}\left( t\right) \) is at most \( n \) . If \( p\left( t\right) \) is any monic polynomial such that \( p\left( A\right) = 0 \), then \... | Proof. The set of monic polynomials that annihilate \( A \) contains \( {p}_{A}\left( t\right) \), which has degree \( n \) . Let \( m = \min \{ k : p\left( t\right) \) is a monic polynomial of degree \( k \) and \( p\left( A\right) = 0\} \) ; necessarily \( m \leq n \) . If \( p\left( t\right) \) is any monic polynomi... | Yes |
Corollary 3.3.3. Similar matrices have the same minimal polynomial. | Proof. If \( A, B, S \in {M}_{n} \) and if \( A = {SB}{S}^{-1} \), then \( {q}_{B}\left( A\right) = {q}_{B}\left( {{SB}{S}^{-1}}\right) = \) \( S{q}_{B}\left( B\right) {S}^{-1} = 0 \), so \( {q}_{B}\left( t\right) \) is a monic polynomial that annihilates \( A \) and hence the degree of \( {q}_{A}\left( t\right) \) is ... | Yes |
Corollary 3.3.4. For each \( A \in {M}_{n} \), the minimal polynomial \( {q}_{A}\left( t\right) \) divides the characteristic polynomial \( {p}_{A}\left( t\right) \) . Moreover, \( {q}_{A}\left( \lambda \right) = 0 \) if and only if \( \lambda \) is an eigenvalue of \( A \) , so every root of \( {p}_{A}\left( t\right) ... | Proof. Since \( {p}_{A}\left( A\right) = 0 \), the fact that there is a polynomial \( h\left( t\right) \) such that \( {p}_{A}\left( t\right) = \) \( h\left( t\right) {q}_{A}\left( t\right) \) follows from (3.2.1). This factorization makes it clear that every root of \( {q}_{A}\left( t\right) = 0 \) is a root of \( {p}... | Yes |
Corollary 3.3.8. Let \( A \in {M}_{n} \) have distinct eigenvalues \( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{d} \) and let\n\n\[ q\left( t\right) = \left( {t - {\lambda }_{1}}\right) \left( {t - {\lambda }_{2}}\right) \cdots \left( {t - {\lambda }_{d}}\right) \]\n\nThen \( A \) is diagonalizable if and only ... | This criterion is actually useful for determining if a given matrix is diagonalizable, provided that we know its distinct eigenvalues: Form the polynomial (3.3.9) and see if it annihilates \( A \) . If it does, it must be the minimal polynomial of \( A \), since no lower-order polynomial could have as zeroes all the di... | Yes |
Theorem 3.5.3. Let \( A \in {M}_{n} \) be given. Then\n\n(a) A has an LU factorization in which \( L \) is nonsingular if and only if \( A \) has the row inclusion property: For each \( i = 1,\ldots, n - 1, A\left\lbrack {\{ i + 1;1,\ldots, i\} }\right\rbrack \) is a linear combination of the rows of \( A\left\lbrack {... | Proof. If \( A = {LU} \), then \( A\left\lbrack {\{ 1,\ldots, i + 1\} }\right\rbrack = L\left\lbrack {\{ 1,\ldots, i + 1\} }\right\rbrack U\left\lbrack {\{ 1,\ldots, i + 1\} }\right\rbrack \) . Thus, to verify the necessity of the row inclusion property, it suffices to take \( i = k = n - 1 \) in the partitioned presen... | Yes |
Lemma 3.5.7. Let \( A \in {M}_{k} \) be nonsingular: Then there is a permutation matrix \( P \in {M}_{k} \) such that \( \det \left( {{P}^{T}A}\right) \left\lbrack {\{ 1,\ldots, j\} }\right\rbrack \neq 0, j = 1,\ldots, k \) . | Proof. The proof is by induction on \( k \) . If \( k = 1 \) or 2, the result is clear by inspection. Suppose that it is valid up to and including \( k - 1 \) . Consider a nonsingular \( A \in {M}_{k} \) and delete its last column. The remaining \( k - 1 \) columns are linearly independent and hence they contain \( k -... | Yes |
Theorem 3.5.8 (PLU factorization). For each \( A \in {M}_{n} \) there is a permutation matrix \( P \in {M}_{n} \), a unit lower triangular \( L \in {M}_{n} \), and an upper triangular \( U \in {M}_{n} \) such that \( A = {PLU} \) . | Proof. If we show that there is a permutation matrix \( Q \) such that \( {QA} \) has the row inclusion property, then (3.5.3) and the exercise following it ensure that \( {QA} = {LU} \) with a unit lower triangular factor \( L \), so \( A = {PLU} \) for \( P = {Q}^{T} \) .\n\nIf \( A \) is nonsingular, the desired per... | Yes |
Theorem 3.5.13. Let \( A, B \in {M}_{n} \) be nonsingular. The following are equivalent:\n\n(a) There is a unique permutation matrix \( P \in {M}_{n} \) such that both \( A \) and \( B \) are triangularly equivalent to \( P \) .\n\n(b) \( A \) and \( B \) are triangularly equivalent.\n\n(c) The rank equalities (3.5.10)... | Proof. The implication (a) \( \Rightarrow \) (b) is clear, and the implication (b) \( \Rightarrow \) (c) is the content of the exercise preceding (3.5.11). If \( A = {L}_{1}P{U}_{1} \) and \( B = {L}_{2}{P}^{\prime }{U}_{2} \) are \( {LPU} \) factorizations and if the hypothesis (c) is assumed, then (using the notation... | No |
Theorem 3.5.14 (LPDU factorization). For each nonsingular \( A \in {M}_{n} \) there is a unique permutation matrix \( P \), a unique nonsingular diagonal matrix \( D \), a unit lower triangular matrix \( L \), and a unit upper triangular matrix \( U \) such that \( A = {LPDU} \) . | Proof. Theorem 3.5.11 ensures that there is a unit lower triangular matrix \( L \), a unique permutation matrix \( P \), and a nonsingular upper triangular matrix \( {U}^{\prime } \) such that \( A = \) \( {LP}{U}^{\prime } \) . Let \( D \) denote the diagonal matrix whose respective diagonal entries are the same as th... | Yes |
If \( f : D \rightarrow \mathbf{R} \) is a twice continuously differentiable function on some domain \( D \subset {\mathbf{R}}^{n} \), the real matrix \[ H\left( x\right) = \left\lbrack {{h}_{ij}\left( x\right) }\right\rbrack = \left\lbrack \frac{{\partial }^{2}f\left( x\right) }{\partial {x}_{i}\partial {x}_{j}}\right... | Thus, the Hessian matrix of a real-valued twice continuously differentiable function is always a real symmetric matrix. | Yes |
Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) have real or complex entries, and consider the quadratic form on \( {\mathbf{R}}^{n} \) or \( {\mathbf{C}}^{n} \) generated by \( A \) : | \[ Q\left( x\right) = {x}^{T}{Ax} = \mathop{\sum }\limits_{{i, j = 1}}^{n}{a}_{ij}{x}_{i}{x}_{j} = \mathop{\sum }\limits_{{i, j = 1}}^{n}\frac{1}{2}\left( {{a}_{ij} + {a}_{ji}}\right) {x}_{i}{x}_{j} = {x}^{T}\left\lbrack {\frac{1}{2}\left( {A + {A}^{T}}\right) }\right\rbrack x \] Thus, \( A \) and \( \frac{1}{2}\left( ... | Yes |
Consider a second-order linear partial differential operator \( L \) defined by\n\n\[ \n{Lf}\left( x\right) = \mathop{\sum }\limits_{{i, j = 1}}^{n}{a}_{ij}\left( x\right) \frac{{\partial }^{2}f\left( x\right) }{\partial {x}_{i}\partial {x}_{j}}\n\]\n\nThe coefficient functions \( {a}_{ij} \) and the function \( f \) a... | \[ \n{Lf} = \mathop{\sum }\limits_{{i, j = 1}}^{n}{a}_{ij}\left( x\right) \frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}} = \mathop{\sum }\limits_{{i, j = 1}}^{n}\frac{1}{2}\left\lbrack {{a}_{ij}\left( x\right) \frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}} + {a}_{ji}\left( x\right) \frac{{\partial... | Yes |
Consider an undirected graph \( \Gamma \) : a collection \( N \) of nodes \( \left\{ {{P}_{1},{P}_{2},\ldots ,{P}_{n}}\right\} \) and a collection \( E \) of unordered pairs of nodes called edges, \( E = \left\{ {\left\{ {{P}_{{i}_{1}},{P}_{{j}_{1}}}\right\} ,\left\{ {{P}_{{i}_{2}},{P}_{{j}_{2}}}\right\} ,\ldots }\righ... | Since \( \Gamma \) is undirected, its adjacency matrix is symmetric. | No |
Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n}\left( \mathbf{R}\right) \) and consider the real bilinear form\n\n\[ Q\left( {x, y}\right) = {y}^{T}{Ax} = \mathop{\sum }\limits_{{i, j = 1}}^{n}{a}_{ij}{y}_{i}{x}_{j},\;x, y \in {\mathbf{R}}^{n} \]\n\nwhich reduces to the ordinary inner product when \( A = I \)... | To show this, it suffices to observe that if \( x = {e}_{j} \) and \( y = {e}_{i} \), then \( Q\left( {{e}_{j},{e}_{i}}\right) = {a}_{ij} \) and \( Q\left( {{e}_{i},{e}_{j}}\right) = {a}_{ji} \) . Thus, symmetric real bilinear forms are naturally associated with symmetric real matrices. | Yes |
Theorem 4.1.2 (Toeplitz decomposition). Each \( A \in {M}_{n} \) can be written uniquely as \( A = H + {iK} \), in which both \( H \) and \( K \) are Hermitian. It can also be written uniquely as \( A = H + S \), in which \( H \) is Hermitian and \( S \) is skew Hermitian. | Proof. Write \( A = \frac{1}{2}\left( {A + {A}^{ * }}\right) + i\left\lbrack {\frac{1}{2i}\left( {A - {A}^{ * }}\right) }\right\rbrack \) and observe that both \( H = \frac{1}{2}\left( {A + {A}^{ * }}\right) \) and \( K = \frac{1}{2i}\left( {A - {A}^{ * }}\right) \) are Hermitian. For the uniqueness assertion, observe ... | Yes |
Theorem 4.1.3. Let \( A \in {M}_{n} \) be Hermitian. Then\n\n(a) \( {x}^{ * }{Ax} \) is real for all \( x \in {\mathbf{C}}^{n} \)\n\n(b) the eigenvalues of \( A \) are real\n\n(c) \( {S}^{ * }{AS} \) is Hermitian for all \( S \in {M}_{n} \) | Proof. Compute \( \left( \overline{{x}^{ * }{Ax}}\right) = {\left( {x}^{ * }Ax\right) }^{ * } = {x}^{ * }{A}^{ * }x = {x}^{ * }{Ax} \), so \( {x}^{ * }{Ax} \) equals its complex conjugate and hence is real. If \( {Ax} = {\lambda x} \) and \( {x}^{ * }x = 1 \), then \( \lambda = \lambda {x}^{ * }x = {x}^{ * }{\lambda x}... | Yes |
Theorem 4.1.4. Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) be given. Then \( A \) is Hermitian if and only if at least one of the following conditions is satisfied:\n\n(a) \( {x}^{ * }{Ax} \) is real for all \( x \in {\mathbf{C}}^{n} \)\n\n(b) \( A \) is normal and has only real eigenvalues\n\n(c) \( {... | Proof. It suffices to prove only the sufficiency of each condition. If \( {x}^{ * }{Ax} \) is real for all \( x \in {\mathbf{C}}^{n} \), then \( {\left( x + y\right) }^{ * }A\left( {x + y}\right) = \left( {{x}^{ * }{Ax} + {y}^{ * }{Ay}}\right) + \left( {{x}^{ * }{Ay} + {y}^{ * }{Ax}}\right) \) is real for all \( x, y \... | Yes |
Theorem 4.1.8. Let \( A \in {M}_{n} \) be given. Then \( {x}^{ * }{Ax} \) is real and positive (respectively, \( {x}^{ * }{Ax} \) is real and nonnegative) for all nonzero \( x \in {\mathbf{C}}^{n} \) if and only if \( A \) is Hermitian and all of its eigenvalues are positive (respectively, nonnegative). | Proof. If \( {x}^{ * }{Ax} \) is real and positive (respectively, real and nonnegative) whenever \( x \neq 0 \) , then \( {x}^{ * }{Ax} \) is real for all \( x \in {\mathbf{C}}^{n} \), so (4.1.4(a)) ensures that \( A \) is Hermitian. Moreover, \( \lambda = {u}^{ * }\left( {\lambda u}\right) = {u}^{ * }{Au} \) if \( u \... | Yes |
Theorem 4.1.10. Let \( A \in {M}_{n}\left( \mathbf{R}\right) \) be symmetric. Then \( {x}^{T}{Ax} > 0 \) (respectively, \( {x}^{T}{Ax} \geq 0 \) ) for all nonzero \( x \in {\mathbf{R}}^{n} \) if and only if every eigenvalue of \( A \) is positive (respectively, nonnegative). | Proof. Since \( A \) is Hermitian, it suffices to show that \( {z}^{ * }{Az} > 0 \) (respectively, \( {z}^{ * }{Az} \geq \) \( 0) \) whenever \( z = x + {iy} \in {\mathbf{C}}^{n} \) with \( x, y \in {\mathbf{R}}^{n} \) and at least one is nonzero. Since \( = \) \( {\left( {y}^{T}Ax\right) }^{T} = {x}^{T}{Ay} \), we hav... | No |
Theorem 4.2.2 (Rayleigh). Let \( A \in {M}_{n} \) be Hermitian, let the eigenvalues of \( A \) be ordered as in (4.2.1), let \( {i}_{1},\ldots ,{i}_{k} \) be given integers with \( 1 \leq {i}_{1} < \cdots < {i}_{k} \leq n \), let \( {x}_{{i}_{1}},\ldots ,{x}_{{i}_{k}} \) be orthonormal and such that \( A{x}_{{i}_{p}} =... | Proof. If \( x \in S \) is nonzero, then \( \xi = x/\parallel x{\parallel }_{2} \) is a unit vector and \( {x}^{ * }{Ax}/{x}^{ * }x = {x}^{ * }{Ax}/ \) \( \parallel x{\parallel }_{2}^{2} = {\xi }^{ * }{A\xi } \) . For any given unit vector \( x \in S \), there are scalars \( {\alpha }_{1},\ldots ,{\alpha }_{k} \) such ... | Yes |
Lemma 4.2.3 (Subspace intersection). Let \( {S}_{1},\ldots ,{S}_{k} \) be given subspaces of \( {\mathbf{C}}^{n} \) . If \( \delta = \dim {S}_{1} + \cdots + \dim {S}_{k} - \left( {k - 1}\right) n \geq 1 \), there are orthonormal vectors \( {x}_{1},\ldots ,{x}_{\delta } \) such that \( {x}_{1},\ldots ,{x}_{\delta } \in ... | Proof. See (0.1.7). The set \( {S}_{1} \cap \cdots \cap {S}_{k} \) is a subspace, and the stated inequality ensures that \( \dim \left( {{S}_{1} \cap \cdots \cap {S}_{k}}\right) \geq \delta \geq 1 \) . Let \( {x}_{1},\ldots ,{x}_{\delta } \) be any \( \delta \) elements of an orthonormal basis of \( {S}_{1} \cap \cdots... | No |
Theorem 4.2.6 (Courant-Fischer). Let \( A \in {M}_{n} \) be Hermitian and let \( {\lambda }_{1} \leq \cdots \leq {\lambda }_{n} \) be its algebraically ordered eigenvalues. Let \( k \in \{ 1,\ldots, n\} \) and let \( S \) denote a subspace of \( {\mathbf{C}}^{n} \) . Then\n\n\[ \n{\lambda }_{k} = \mathop{\min }\limits_... | Proof. Let \( {x}_{1},\ldots ,{x}_{n} \in {\mathbf{C}}^{n} \) be orthonormal and such that \( A{x}_{i} = {\lambda }_{i}{x}_{i} \) for each \( i = \) \( 1,\ldots, n \) . Let \( S \) be any \( k \) -dimensional subspace of \( {\mathbf{C}}^{n} \) and let \( {S}^{\prime } = \operatorname{span}\left\{ {{x}_{k},\ldots ,{x}_{... | Yes |
Theorem 4.2.10. Let \( A \in {M}_{n} \) be Hermitian, let the eigenvalues of \( A \) be arranged in increasing order (4.2.1), let \( S \) be a given \( k \) -dimensional subspace of \( {\mathbf{C}}^{n} \), and let \( c \in \mathbf{R} \) be given.\n\n(a) If \( {x}^{ * }{Ax} \geq c \) (respectively, \( {x}^{ * }{Ax} > c ... | Proof. Let \( {x}_{1},\ldots ,{x}_{n} \in {\mathbf{C}}^{n} \) be orthonormal and such that \( A{x}_{i} = {\lambda }_{i}\left( A\right) {x}_{i} \) for each \( i = 1,\ldots, n \) and let \( {S}_{1} = \operatorname{span}\left\{ {{x}_{1},\ldots ,{x}_{n - k + 1}}\right\} \) . Then \( \dim S + \dim {S}_{1} = k + (n - k + \) ... | Yes |
Corollary 4.2.12. Let \( A \in {M}_{n} \) be Hermitian. If \( {x}^{ * }{Ax} \geq 0 \) for all \( x \) in a \( k \) -dimensional subspace, then \( A \) has at least \( k \) nonnegative eigenvalues. If \( {x}^{ * }{Ax} > 0 \) for all nonzero \( x \) in a \( k \) -dimensional subspace, then \( A \) has at least \( k \) po... | Proof. The preceding theorem ensures that \( {\lambda }_{n - k + 1}\left( A\right) \geq 0 \) (respectively, \( {\lambda }_{n - k + 1}\left( A\right) > \) \( 0) \), and \( {\lambda }_{n}\left( A\right) \geq \cdots \geq {\lambda }_{n - k + 1}\left( A\right) \) . | Yes |
Theorem 4.3.1 (Weyl). Let \( A, B \in {M}_{n} \) be Hermitian and let the respective eigenvalues of \( A, B \), and \( A + B \) be \( {\left\{ {\lambda }_{i}\left( A\right) \right\} }_{i = 1}^{n},{\left\{ {\lambda }_{i}\left( B\right) \right\} }_{i = 1}^{n} \), and \( {\left\{ {\lambda }_{i}\left( A + B\right) \right\}... | Proof. Let \( {x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n} \), and \( {z}_{1},\ldots ,{z}_{n} \) be orthonormal lists of eigenvectors of \( A, B \), and \( A + B \), respectively, such that \( A{x}_{i} = {\lambda }_{i}\left( A\right) {x}_{i}, B{y}_{i} = {\lambda }_{i}\left( B\right) {y}_{i} \) , and \( \left( {A + B... | Yes |
Corollary 4.3.3. Let \( A, B \in {M}_{n} \) be Hermitian. Suppose that \( B \) has exactly \( \pi \) positive eigenvalues and exactly \( v \) negative eigenvalues. Then\n\n\[{\lambda }_{i}\left( {A + B}\right) \leq {\lambda }_{i + \pi }\left( A\right) ,\;i = 1,\ldots, n - \pi\]\n\n(4.3.4a)\n\nwith equality for some \( ... | Proof. Take \( j = n - \pi \) in (4.3.2a) and use the preceding exercise to obtain \( {\lambda }_{i}(A + \) \( B) \leq {\lambda }_{i + \pi }\left( A\right) + {\lambda }_{n - \pi }\left( B\right) \leq {\lambda }_{i + \pi }\left( A\right) \) with equality if and only if \( B \) is singular and there is a nonzero vector \... | Yes |
Corollary 4.3.5. Let \( A, B \in {M}_{n} \) be Hermitian. Suppose that \( B \) is singular and \( \operatorname{rank}B = r \) . Then\n\n\[ \n{\lambda }_{i}\left( {A + B}\right) \leq {\lambda }_{i + r}\left( A\right) ,\;i = 1,\ldots, n - r \n\]\n\n(4.3.6a)\n\nwith equality for some \( i \) if and only if \( {\lambda }_{... | Proof. To verify (4.3.6a), take \( j = r \) in (4.3.2a) and use the preceding exercise to obtain \( {\lambda }_{i}\left( {A + B}\right) \leq {\lambda }_{i + r}\left( A\right) + {\lambda }_{n - r}\left( B\right) \leq {\lambda }_{i + r}\left( A\right) \) with equality if and only if \( {\lambda }_{n - r}\left( B\right) =... | Yes |
Corollary 4.3.7. Let \( A, B \in {M}_{n} \) be Hermitian. Suppose that \( B \) has exactly one positive eigenvalue and exactly one negative eigenvalue. Then\n\n\[ \n{\lambda }_{1}\left( {A + B}\right) \leq {\lambda }_{2}\left( A\right) \]\n\n\[ \n{\lambda }_{i - 1}\left( A\right) \leq {\lambda }_{i}\left( {A + B}\right... | Proof. Take \( \pi = v = 1 \) in (4.3.4a, b) and use the preceding exercise. | No |
Corollary 4.3.9. Let \( n \geq 2 \), let \( A \in {M}_{n} \) be Hermitian, and let \( z \in {\mathbf{C}}^{n} \) be nonzero. Then\n\n\[ \n{\lambda }_{i}\left( A\right) \leq {\lambda }_{i}\left( {A + z{z}^{ * }}\right) \leq {\lambda }_{i + 1}\left( A\right) ,\;i = 1,\ldots, n - 1 \n\] \n\n(4.3.10) \n\n\[ \n{\lambda }_{n}... | Proof. In (4.3.4a), take \( \pi = 1 \) and \( v = 0 \) ; in (4.3.4b), take \( \pi = 0 \) and \( v = 1 \) . Use the preceding exercise. | No |
Corollary 4.3.12. Let \( A, B \in {M}_{n} \) be Hermitian and suppose that \( B \) is positive semidefinite. Then\n\n\[{\lambda }_{i}\left( A\right) \leq {\lambda }_{i}\left( {A + B}\right) ,\;i = 1,\ldots, n\]\n\n(4.3.13)\n\nwith equality for some \( i \) if and only if \( B \) is singular and there is a nonzero vecto... | Proof. Use (4.3.4b) with \( v = 0 \) and use the preceding exercise. If \( B \) is nonsingular, equality cannot occur in (4.3.13). | No |
Corollary 4.3.15. Let \( A, B \in {M}_{n} \) be Hermitian. Then\n\n\[ \n{\lambda }_{i}\left( A\right) + {\lambda }_{1}\left( B\right) \leq {\lambda }_{i}\left( {A + B}\right) \leq {\lambda }_{i}\left( A\right) + {\lambda }_{n}\left( B\right) ,\;i = 1,\ldots, n \n\] \n\n(4.3.16) \n\nwith equality in the upper bound if a... | Proof. Take \( j = 0 \) in (4.3.2a) and \( j = 1 \) in (4.3.2b). | Yes |
Theorem 4.3.17 (Cauchy). Let \( B \in {M}_{n} \) be Hermitian, let \( y \in {\mathbf{C}}^{n} \) and \( a \in \mathbf{R} \) be a given, and let \( A = \left\lbrack \begin{matrix} B & y \\ {y}^{ * } & a \end{matrix}\right\rbrack \in {M}_{n + 1} \) . Then\n\n\[ \n{\lambda }_{1}\left( A\right) \leq {\lambda }_{1}\left( B\r... | Proof. The asserted interlacing of ordered eigenvalues is unchanged if we replace \( A \) with \( A + \mu {I}_{n + 1} \), which replaces \( B \) with \( B + \mu {I}_{n} \) . Thus, there is no loss of generality to assume that \( B \) and \( A \) are positive definite. Consider the Hermitian matrices \( \mathcal{H} = \l... | Yes |
Let \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) and \( {\mu }_{1},\ldots ,{\mu }_{n} \) be real numbers that satisfy the interlacing inequalities\n\n\[ \n{\lambda }_{1} \leq {\mu }_{1} \leq {\lambda }_{2} \leq {\mu }_{2} \leq \cdots \leq {\lambda }_{n} \leq {\mu }_{n} \n\]\n\nLet \( \Lambda = \operatorname{diag}\left( ... | Proof. There is no loss of generality to assume that \( {\lambda }_{1} > 0 \), for if \( {\lambda }_{1} \leq 0 \) let \( c > - {\lambda }_{1} \) and replace each \( {\lambda }_{i} \) by \( {\lambda }_{i} + c \) and each \( {\mu }_{i} \) by \( {\mu }_{i} + c \). This shift does not disturb the interlacing inequalities (... | Yes |
Theorem 4.3.28. Let \( A \in {M}_{n} \) be Hermitian, partitioned as\n\n\[ A = \left\lbrack \begin{matrix} B & C \\ {C}^{ * } & D \end{matrix}\right\rbrack ,\;B \in {M}_{m}, D \in {M}_{n - m}, C \in {M}_{m, n - m} \]\n\nLet the eigenvalues of \( A \) and \( B \) be ordered as in (4.2.1). Then\n\n\[ {\lambda }_{i}\left(... | Proof. Let \( {x}_{1},\ldots ,{x}_{n} \in {\mathbf{C}}^{n} \) and \( {y}_{1},\ldots ,{y}_{n} \in {\mathbf{C}}^{m} \) be orthonormal lists of eigenvectors of \( A \) and \( B \), respectively, such that \( A{x}_{i} = {\lambda }_{i}\left( A\right) {x}_{i} \) for each \( i = 1,\ldots, n \) and \( B{y}_{i} = \) \( {\lambda... | Yes |
Corollary 4.3.37. Let \( A \in {M}_{n} \) be Hermitian, suppose that \( 1 \leq m \leq n \), and let \( {u}_{1},\ldots ,{u}_{m} \in {\mathbf{C}}^{n} \) be orthonormal. Let \( {B}_{m} = {\left\lbrack {u}_{i}^{ * }A{u}_{j}\right\rbrack }_{i, j = 1}^{m} \in {M}_{m} \) and let the eigenvalues of \( A \) and \( {B}_{m} \) be... | Proof. If \( m < n \), choose \( n - m \) additional vectors \( {u}_{m + 1},\ldots ,{u}_{n} \) so that \( U = \) \( \left\lbrack \begin{array}{lll} {u}_{1} & \ldots & {u}_{n} \end{array}\right\rbrack \in {M}_{n} \) is unitary. Then \( {U}^{ * }{AU} \) has the same eigenvalues as \( A \), and \( {B}_{m} \) is a principa... | Yes |
Corollary 4.3.39. Let \( A \in {M}_{n} \) be Hermitian and suppose that \( 1 \leq m \leq n \) . Then\n\n\[ \n{\lambda }_{1}\left( A\right) + \cdots + {\lambda }_{m}\left( A\right) = \mathop{\min }\limits_{\substack{{V \in {M}_{n, m}} \\ {{V}^{ * }V = {I}_{m}} }}\operatorname{tr}{V}^{ * }{AV} \n\] | For each \( m = 1,\ldots, n - 1 \) the minimum or maximum in (4.3.40) is achieved for a matrix \( V \) whose columns are orthonormal eigenvectors associated with the \( m \) smallest or largest eigenvalues of \( A \) ; for \( m = n \) we have \( \operatorname{tr}{V}^{ * }{AV} = \operatorname{tr}{AV}{V}^{ * } = \operato... | No |
Theorem 4.3.45 (Schur). Let \( A = \left\lbrack {a}_{ij}\right\rbrack \in {M}_{n} \) be Hermitian. Its vector of eigenvalues \( \lambda \left( A\right) = {\left\lbrack {\lambda }_{i}\left( A\right) \right\rbrack }_{i = 1}^{n} \) majorizes its vector of main diagonal entries \( d\left( A\right) = {\left\lbrack {a}_{ii}\... | Proof. Let \( P \in {M}_{n} \) be a permutation matrix such that the \( i, i \) entry of \( {PA}{P}^{T} \) is \( {d}_{i}{\left( A\right) }^{ \downarrow } \) for each \( i = 1,\ldots, n \) (0.9.5). The vector of eigenvalues \( {PA}{P}^{T} \) (as well as of \( A \) ) is \( \lambda \left( A\right) \) of \( A \) . Partitio... | Yes |
Theorem 4.3.47. Let \( A, B \in {M}_{n} \) be Hermitian. Let \( \lambda \left( A\right) ,\lambda \left( B\right) \), and \( \lambda \left( {A + B}\right) \) , respectively, denote the real \( n \) -vectors of eigenvalues of \( A, B \), and \( A + B \), respectively. Then\n\n(a) (Fan) \( \lambda {\left( A\right) }^{ \do... | Proof. (a) For any \( k \in \{ 1,\ldots, n - 1\} \), use (4.3.40) to write the sum of the \( k \) largest eigenvalues of \( A + B \) as\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}{\left( A + B\right) }^{ \downarrow } = \mathop{\max }\limits_{\substack{{V \in {M}_{n, k}} \\ {{V}^{ * }V = {I}_{k}} }}\operator... | Yes |
Theorem 4.3.48. Let \( n \geq 1 \), let \( x = \left\lbrack {x}_{i}\right\rbrack \in {\mathbf{R}}^{n} \) and \( y = \left\lbrack {y}_{i}\right\rbrack \in {\mathbf{R}}^{n} \) be given, and suppose that \( x \) majorizes \( y \) . Let \( \Lambda = \operatorname{diag}x \in {M}_{n}\left( \mathbf{R}\right) \) . There exists... | Proof. There is no loss of generality if we assume that the entries of the vectors \( x \) and \( y \) are in nonincreasing order: \( {x}_{1} \geq {x}_{2} \geq \cdots \) and \( {y}_{1} \geq {y}_{2} \geq \cdots \) . The assertion is trivial for \( n = 1 : {x}_{1} = {y}_{1}, Q = \left\lbrack 1\right\rbrack \), and \( A =... | Yes |
Theorem 4.3.50. Let \( x = \left\lbrack {x}_{i}\right\rbrack \in {\mathbf{R}}^{n} \) and \( z = \left\lbrack {z}_{i}\right\rbrack \in {\mathbf{C}}^{n} \) be given. Then \( x \) majorizes \( \operatorname{Re}z = {\left\lbrack \operatorname{Re}{z}_{i}\right\rbrack }_{i = 1}^{n} \) if and only if there is an \( A \in {M}_... | Proof. Let \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) be the eigenvalues of \( A \in {M}_{n} \), and use (2.3.1) to write \( A = \) \( {UT}{U}^{ * } \), in which \( T = \left\lbrack {t}_{ij}\right\rbrack \in {M}_{n} \) is upper triangular and \( {t}_{ii} = {\lambda }_{i} \) for \( i = 1,\ldots, n \) . A computation re... | Yes |
Lemma 4.3.51. Let \( x = \left\lbrack {x}_{i}\right\rbrack, y = \left\lbrack {y}_{i}\right\rbrack, w = \left\lbrack {w}_{i}\right\rbrack \in {\mathbf{R}}^{n} \). Suppose that \( x \) majorizes \( y \). Then\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{w}_{i}^{ \downarrow }{x}_{i}^{ \uparrow } \leq \mathop{\sum }\limits_{{... | Proof. Since \( x \) majorizes \( y \), we have \( {\widehat{X}}_{k} \geq {Y}_{k} \) for each \( k = 1,\ldots, n - 1 \) and \( {\widehat{X}}_{n} = {Y}_{n} \). Use partial summation (twice) and the assumption that each \( {w}_{i}^{ \downarrow } - {w}_{i + 1}^{ \downarrow } \) is nonnegative to compute\n\n\[ \mathop{\sum... | Yes |
Theorem 4.3.53. Let \( A, B \in {M}_{n} \) be Hermitian and have respective vectors of eigenvalues \( \lambda \left( A\right) = {\left\lbrack {\lambda }_{i}\left( A\right) \right\rbrack }_{i = 1}^{n} \) and \( \lambda \left( B\right) = {\left\lbrack {\lambda }_{i}\left( B\right) \right\rbrack }_{i = 1}^{n} \). Then\n\n... | Proof. Let \( A = {U\Lambda }{U}^{ * } \), in which \( U \in {M}_{n} \) is unitary and \( \Lambda = \operatorname{diag}\lambda {\left( A\right) }^{ \downarrow } \). Let \( \widetilde{B} = \left\lbrack {\beta }_{ij}\right\rbrack = {U}^{ * }{BU} \). Then \( \operatorname{tr}{AB} = \operatorname{tr}{U\Lambda }{U}^{ * }B =... | Yes |
Lemma 4.4.2. Let \( A \in {M}_{n} \) be given, let \( \lambda \) be an eigenvalue of \( A\bar{A} \), and let \( x \in {\mathbf{C}}^{n} \) be a unit eigenvector of \( A\bar{A} \) associated with \( \lambda \). Let \( \mathcal{S} = \operatorname{span}\{ A\bar{x}, x\} \), which has dimension one or two.\n\n(a) If \( \dim ... | Proof. (a) If \( \dim \mathcal{S} = 1 \), then \( \{ A\bar{x}, x\} \) is linearly dependent and \( A\bar{x} = {\mu x} \) for some \( \mu \in \mathbf{C} \). Compute \( {\lambda x} = A\bar{A}x = A\left( \overline{A\bar{x}}\right) = A\bar{\mu }\bar{x} = \bar{\mu }\bar{A}\bar{x} = \bar{\mu }{\mu x} = {\left| \mu \right| }^... | Yes |
Theorem 4.4.3. Let \( A \in {M}_{n} \) and \( p \in \{ 0,1,\ldots, n\} \) be given. Suppose that \( A\bar{A} \) has at least \( p \) real nonnegative eigenvalues, including \( {\lambda }_{1},\ldots ,{\lambda }_{p} \) . Then there is a unitary \( U \in {M}_{n} \) such that\n\n\[ A = U\left\lbrack \begin{matrix} \Delta &... | Proof. The case \( n = 1 \) is trivial (see the preceding exercise), as is the case \( p = 0 \), so we assume that \( n \geq 2 \) and \( p \geq 1 \) .\n\nConsider the following reduction: Let \( x \) be a unit eigenvector of \( A\bar{A} \) associated with a real nonnegative eigenvalue \( \lambda \) and let \( \sigma = ... | Yes |
Corollary 4.4.4. Let \( A \in {M}_{n} \) be given.\n\n(a) If there is a unitary \( U \in {M}_{n} \) such that \( A = {U\Delta }{U}^{T} \) and \( \Delta \) is upper triangular, then every eigenvalue of \( A\bar{A} \) is nonnegative. | Proof. (a) If \( A = {U\Delta }{U}^{T} \) and \( \Delta \) is upper triangular, then the eigenvalues of \( A\bar{A} = \) \( {U\Delta }{U}^{T}\bar{U}\bar{\Delta }{U}^{ * } = {U\Delta }\bar{\Delta }{U}^{ * } \) are the main diagonal entries of \( \Delta \bar{\Delta } \), which are nonnegative. | Yes |
Proposition 4.4.5. Let \( A \in {M}_{2} \) be given and let \( {\sigma }_{1} \geq {\sigma }_{2} \geq 0 \) be the singular values of \( S\left( A\right) = \frac{1}{2}\left( {A + {A}^{T}}\right) \), the symmetric part of \( A \) . If \( {\sigma }_{1} = {\sigma }_{2} \), let \( \sigma = {\sigma }_{1} \) .\n\n(a) \( A \) i... | Proof. (a) Write \( A = S\left( A\right) + C\left( A\right) \) as the sum of its symmetric and skew-symmetric parts (0.2.5). The preceding corollary ensures that there is a unitary \( U \in {M}_{2} \) such that \( S\left( A\right) = U\left\lbrack \begin{matrix} {\sigma }_{1} & 0 \\ 0 & {\sigma }_{2} \end{matrix}\right\... | Yes |
Theorem 4.4.9 (Youla). Let \( A \in {M}_{n} \) be given. Let \( p \in \{ 0,1,\ldots, n\} \) and suppose that \( A\bar{A} \) has exactly preal nonnegative eigenvalues. Then there is a unitary \( U \in {M}_{n} \) such that \[ A = U\left\lbrack \begin{matrix} \Delta & \star \\ 0 & \Gamma \end{matrix}\right\rbrack {U}^{T} ... | Proof. Theorem 4.4.3 ensures that \( A \) is unitarily congruent to a block upper triangular matrix of the form \( \left\lbrack \begin{matrix} \Delta & \star \\ 0 & C \end{matrix}\right\rbrack \), in which \( \Delta \in {M}_{p} \) has the stated properties. It suffices to consider a matrix \( C \in {M}_{q} \) such that... | Yes |
Corollary 4.4.13. Let \( A \in {M}_{n} \) be given. The non-real eigenvalues of \( A\bar{A} \) occur in conjugate pairs. The real negative eigenvalues of \( A\bar{A} \) occur in equal pairs. | Proof. Factor \( A \) as in (4.4.10). The eigenvalues of \( A\bar{A} \) are the eigenvalues of \( \Delta \bar{\Delta } \oplus \Gamma \bar{\Gamma } \) , which are the squares of the absolute values of the main diagonal entries of \( \Delta \) together with the eigenvalues of all the 2-by-2 diagonal blocks of \( \Gamma \... | Yes |
Lemma 4.4.15. Suppose that \( A \in {M}_{n} \) is partitioned as\n\n\[ A = \left\lbrack \begin{matrix} {A}_{11} & {A}_{12} \\ 0 & {A}_{22} \end{matrix}\right\rbrack \]\n\nin which \( {A}_{11} \) and \( {A}_{22} \) are square. Then \( A \) is conjugate normal if and only if \( {A}_{11} \) and \( {A}_{22} \) are conjugat... | Proof. Proceed as in the proof of (2.5.2) by equating the 1,1 blocks of the identity \( \overline{{A}^{ * }A} = A{A}^{ * } : \overline{{A}_{11}^{ * }{A}_{11}} = {A}_{11}{A}_{11}^{ * } + {A}_{12}{A}_{12}^{ * } \) . However, \( \operatorname{tr}\overline{{A}_{11}^{ * }{A}_{11}} = \overline{\operatorname{tr}{A}_{11}^{ * }... | Yes |
Theorem 4.4.16. A matrix \( A \in {M}_{n} \) is conjugate normal if and only if it is unitarily congruent to a direct sum of the form\n\n\[ \sum \oplus {\tau }_{1}\left\lbrack \begin{matrix} {a}_{1} & {b}_{1} \\ - {b}_{1} & {a}_{1} \end{matrix}\right\rbrack \oplus \cdots \oplus {\tau }_{q}\left\lbrack \begin{matrix} {a... | Proof. Suppose that \( A \) is conjugate normal and factor it as in (4.4.10). Unitary congruence invariance of conjugate normality ensures that \( \left\lbrack \begin{matrix} \Delta & \star \\ 0 & \Gamma \end{matrix}\right\rbrack \) is conjugate normal, and (4.4.15) tells us that it is block diagonal, \( \Delta \) is d... | Yes |
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