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Proposition 16.6. For every matrix \( A \in \mathfrak{{su}}\left( 2\right) \), with\n\n\[ A = \left( \begin{matrix} i{u}_{1} & {u}_{2} + i{u}_{3} \\ - {u}_{2} + i{u}_{3} & - i{u}_{1} \end{matrix}\right) \]\n\nif we write \( \theta = \sqrt{{u}_{1}^{2} + {u}_{2}^{2} + {u}_{3}^{2}} \), then\n\n\[ {e}^{A} = \cos {\theta I}... | Therefore, by the discussion at the end of the previous section, \( {e}^{A} \) is a unit quaternion representing the rotation of angle \( {2\theta } \) and axis \( \left( {{u}_{1},{u}_{2},{u}_{3}}\right) \) (or \( I \) when \( \theta = {k\pi }, k \in \mathbb{Z} \) ). The above formula shows that we may assume that \( 0... | Yes |
Proposition 16.8. Any section \( s : \mathrm{{SO}}\left( 3\right) \rightarrow \mathrm{{SU}}\left( 2\right) \) of \( \rho \) is neither a homomorphism nor continuous. | Proof. Let \( \Gamma \) be the subgroup of \( \mathbf{{SU}}\left( 2\right) \) consisting of all quaternions of the form \( q = \) \( \left\lbrack {a,\left( {b,0,0}\right) }\right\rbrack \) . Then, using the formula for the rotation matrix \( {R}_{q} \) corresponding to \( q \) (and the fact that \( {a}^{2} + {b}^{2} = ... | Yes |
Describe geometrically the rotations defined by the following quaternions:\n\n\[ p = \left( {0, i}\right) ,\;q = \left( {0, j}\right) . \] | Prove that the interpolant \( Z\left( \lambda \right) = p{\left( {p}^{-1}q\right) }^{\lambda } \) is given by\n\n\[ Z\left( \lambda \right) = \left( {0,\cos \left( {{\lambda \pi }/2}\right) i + \sin \left( {{\lambda \pi }/2}\right) j}\right) . \] | No |
Proposition 17.1. Given a Euclidean space \( E \), if \( f : E \rightarrow E \) is a normal linear map, then \( \operatorname{Ker}f = \operatorname{Ker}{f}^{ * } \) . | Proof. First let us prove that\n\n\[ \langle f\left( u\right), f\left( v\right) \rangle = \left\langle {{f}^{ * }\left( u\right) ,{f}^{ * }\left( v\right) }\right\rangle \]\n\nfor all \( u, v \in E \) . Since \( {f}^{ * } \) is the adjoint of \( f \) and \( f \circ {f}^{ * } = {f}^{ * } \circ f \), we have\n\n\[ \langl... | Yes |
Proposition 17.2. Given a Hermitian space \( E \), for any normal linear map \( f : E \rightarrow E \), we have \( \operatorname{Ker}\left( f\right) \cap \operatorname{Im}\left( f\right) = \left( 0\right) \) . | Proof. Assume \( v \in \operatorname{Ker}\left( f\right) \cap \operatorname{Im}\left( f\right) = \left( 0\right) \), which means that \( v = f\left( u\right) \) for some \( u \in E \), and \( f\left( v\right) = 0 \) . By Proposition 17.1, \( \operatorname{Ker}\left( f\right) = \operatorname{Ker}\left( {f}^{ * }\right) ... | Yes |
Proposition 17.4. Given a Hermitian space \( E \), for any normal linear map \( f : E \rightarrow E \), if \( u \) and \( v \) are eigenvectors of \( f \) associated with the eigenvalues \( \lambda \) and \( \mu \) (in \( \mathbb{C} \) ) where \( \lambda \neq \mu \) , then \( \langle u, v\rangle = 0 \) . | Proof. Let us compute \( \langle f\left( u\right), v\rangle \) in two different ways. Since \( v \) is an eigenvector of \( f \) for \( \mu \) , by Proposition 17.3, \( v \) is also an eigenvector of \( {f}^{ * } \) for \( \bar{\mu } \), and we have\n\n\[ \langle f\left( u\right), v\rangle = \langle {\lambda u}, v\rang... | Yes |
Proposition 17.5. Given a Hermitian space \( E \) , all the eigenvalues of any self-adjoint linear map \( f : E \rightarrow E \) are real. | Proof. Let \( z \) (in \( \mathbb{C} \) ) be an eigenvalue of \( f \) and let \( u \) be an eigenvector for \( z \) . We compute \( \langle f\left( u\right), u\rangle \) in two different ways. We have\n\n\[ \langle f\left( u\right), u\rangle = \langle {zu}, u\rangle = z\langle u, u\rangle \]\n\nand since \( f = {f}^{ *... | Yes |
Given a Euclidean space \( E \), if \( f : E \rightarrow E \) is any self-adjoint linear map, then every eigenvalue \( \lambda \) of \( {f}_{\mathbb{C}} \) is real and is actually an eigenvalue of \( f \) (which means that there is some real eigenvector \( u \in E \) such that \( f\left( u\right) = {\lambda u} \) ). Th... | Let \( {E}_{\mathbb{C}} \) be the complexification of \( E,\langle - , - {\rangle }_{\mathbb{C}} \) the complexification of the inner product \( \langle - , - \rangle \) on \( E \), and \( {f}_{\mathbb{C}} : {E}_{\mathbb{C}} \rightarrow {E}_{\mathbb{C}} \) the complexification of \( f : E \rightarrow E \) . By definiti... | Yes |
Given a Hermitian space \( E \), for any linear map \( f : E \rightarrow E \), if \( f \) is skew-self-adjoint, then \( f \) has eigenvalues that are pure imaginary or zero, and if \( f \) is unitary, then \( f \) has eigenvalues of absolute value 1 . | If \( f \) is skew-self-adjoint, \( {f}^{ * } = - f \), and then by the definition of the adjoint map, for any eigenvalue \( \lambda \) and any eigenvector \( u \) associated with \( \lambda \), we have\n\n\[ \lambda \langle u, u\rangle = \langle {\lambda u}, u\rangle = \langle f\left( u\right), u\rangle = \left\langle... | Yes |
Theorem 17.8. (Spectral theorem for self-adjoint linear maps on a Euclidean space) Given a Euclidean space \( E \) of dimension \( n \), for every self-adjoint linear map \( f : E \rightarrow E \), there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of eigenvectors of \( f \) such that the matri... | Proof. We proceed by induction on the dimension \( n \) of \( E \) as follows. If \( n = 1 \), the result is trivial. Assume now that \( n \geq 2 \) . From Proposition 17.6, all the eigenvalues of \( f \) are real, so pick some eigenvalue \( \lambda \in \mathbb{R} \), and let \( w \) be some eigenvector for \( \lambda ... | Yes |
Proposition 17.9. Given a Hermitian space \( E \), for any linear map \( f : E \rightarrow E \) and any subspace \( W \) of \( E \), if \( f\left( W\right) \subseteq W \), then \( {f}^{ * }\left( {W}^{ \bot }\right) \subseteq {W}^{ \bot } \) . Consequently, if \( f\left( W\right) \subseteq W \) and \( {f}^{ * }\left( W... | Proof. If \( u \in {W}^{ \bot } \), then\n\n\[ \langle w, u\rangle = 0\;\text{ for all }w \in W. \]\n\nHowever,\n\n\[ \langle f\left( w\right), u\rangle = \left\langle {w,{f}^{ * }\left( u\right) }\right\rangle \]\n\nand \( f\left( W\right) \subseteq W \) implies that \( f\left( w\right) \in W \) . Since \( u \in {W}^{... | Yes |
Proposition 17.10. If \( f : E \rightarrow E \) is a linear map and \( w = u + {iv} \) is an eigenvector of \( {f}_{\mathbb{C}} : {E}_{\mathbb{C}} \rightarrow {E}_{\mathbb{C}} \) for the eigenvalue \( z = \lambda + {i\mu } \), where \( u, v \in E \) and \( \lambda ,\mu \in \mathbb{R} \), then\n\n\[ f\left( u\right) = {... | Proof. Since\n\n\[ {f}_{\mathbb{C}}\left( {u + {iv}}\right) = f\left( u\right) + {if}\left( v\right) \]\n\nand\n\n\[ {f}_{\mathbb{C}}\left( {u + {iv}}\right) = \left( {\lambda + {i\mu }}\right) \left( {u + {iv}}\right) = {\lambda u} - {\mu v} + i\left( {{\mu u} + {\lambda v}}\right) ,\]\n\nwe have\n\n\[ f\left( u\right... | Yes |
Theorem 17.12. (Main spectral theorem) Given a Euclidean space \( E \) of dimension \( n \), for every normal linear map \( f : E \rightarrow E \), there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) such that the matrix of \( f \) w.r.t. this basis is a block diagonal matrix of the form\n\n\[ \... | Proof. We proceed by induction on the dimension \( n \) of \( E \) as follows. If \( n = 1 \), the result is trivial. Assume now that \( n \geq 2 \) . First, since \( \mathbb{C} \) is algebraically closed (i.e., every polynomial has a root in \( \mathbb{C} \) ), the linear map \( {f}_{\mathbb{C}} : {E}_{\mathbb{C}} \ri... | Yes |
Theorem 17.13. (Spectral theorem for normal linear maps on a Hermitian space) Given a Hermitian space \( E \) of dimension \( n \), for every normal linear map \( f : E \rightarrow E \) there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of eigenvectors of \( f \) such that the matrix of \( f \)... | Proof. We proceed by induction on the dimension \( n \) of \( E \) as follows. If \( n = 1 \), the result is trivial. Assume now that \( n \geq 2 \) . Since \( \mathbb{C} \) is algebraically closed (i.e., every polynomial has a root in \( \mathbb{C} \) ), the linear map \( f : E \rightarrow E \) has some eigenvalue \( ... | Yes |
Theorem 17.14. Given a Euclidean space \( E \) of dimension \( n \) , for every self-adjoint linear map \( f : E \rightarrow E \), there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of eigenvectors of \( f \) such that the matrix of \( f \) w.r.t. this basis is a diagonal matrix\n\n\[\\left( \\... | Proof. We already proved this; see Theorem 17.8. However, it is instructive to give a more direct method not involving the complexification of \( \\langle - , - \\rangle \) and Proposition 17.5.\n\nSince \( \\mathbb{C} \) is algebraically closed, \( {f}_{\\mathbb{C}} \) has some eigenvalue \( \\lambda + {i\\mu } \), an... | Yes |
Given a Euclidean space \( E \) of dimension \( n \), for every skew-self-adjoint linear map \( f : E \rightarrow E \) there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) such that the matrix of \( f \) w.r.t. this basis is a block diagonal matrix of the form \[ \left( \begin{matrix} {A}_{1} & &... | The case where \( n = 1 \) is trivial. As in the proof of Theorem 17.12, \( {f}_{\mathbb{C}} \) has some eigenvalue \( z = \lambda + {i\mu } \), where \( \lambda ,\mu \in \mathbb{R} \) . We claim that \( \lambda = 0 \) . First we show that \[ \langle f\left( w\right), w\rangle = 0 \] for all \( w \in E \) . Indeed, sin... | Yes |
Theorem 17.16. Given a Euclidean space \( E \) of dimension \( n \), for every orthogonal linear map \( f : E \rightarrow E \) there is an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) such that the matrix of \( f \) w.r.t. this basis is a block diagonal matrix of the form\n\n\[ \left( \begin{matrix} ... | Proof. The case where \( n = 1 \) is trivial. It is immediately verified that \( f \circ {f}^{ * } = {f}^{ * } \circ f = \mathrm{{id}} \) implies that \( {f}_{\mathbb{C}} \circ {f}_{\mathbb{C}}^{ * } = {f}_{\mathbb{C}}^{ * } \circ {f}_{\mathbb{C}} = \mathrm{{id}} \), so the map \( {f}_{\mathbb{C}} \) is unitary. By Pro... | Yes |
Theorem 17.17. Let \( E \) be a Euclidean space of dimension \( n \geq 2 \) . For every isometry \( f \in \mathbf{O}\left( E\right) \), if \( p = \dim \left( {E\left( {1, f}\right) }\right) = \dim \left( {\operatorname{Ker}\left( {f - \mathrm{{id}}}\right) }\right) \), then \( f \) is the composition of \( n - p \) ref... | Proof. From Theorem 17.16 there are \( r \) subspaces \( {F}_{1},\ldots ,{F}_{r} \), each of dimension 2, such that\n\n\[ E = E\left( {1, f}\right) \oplus E\left( {-1, f}\right) \oplus {F}_{1} \oplus \cdots \oplus {F}_{r}, \]\n\nand all the summands are pairwise orthogonal. Furthermore, the restriction \( {r}_{i} \) of... | Yes |
Proposition 17.23. (Rayleigh-Ritz) If \( A \) is a symmetric \( n \times n \) matrix with eigenvalues \( {\lambda }_{1} \leq {\lambda }_{2} \leq \cdots \leq {\lambda }_{n} \) and if \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) is any orthonormal basis of eigenvectors of \( A \), where \( {u}_{i} \) is a unit eigenvect... | Proof. First observe that\n\n\[ \mathop{\max }\limits_{{x \neq 0}}\frac{{x}^{\top }{Ax}}{{x}^{\top }x} = \mathop{\max }\limits_{x}\left\{ {{x}^{\top }{Ax} \mid {x}^{\top }x = 1}\right\} \]\n\nand similarly,\n\n\[ \mathop{\max }\limits_{{x \neq 0, x \in {\left\{ {u}_{n - k + 1},\ldots ,{u}_{n}\right\} }^{ \bot }}}\frac{... | Yes |
Proposition 17.24. (Rayleigh-Ritz) If \( A \) is a symmetric \( n \times n \) matrix with eigenvalues \( {\lambda }_{1} \leq {\lambda }_{2} \leq \cdots \leq {\lambda }_{n} \) and if \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) is any orthonormal basis of eigenvectors of \( A \), where \( {u}_{i} \) is a unit eigenvect... | Propositions 17.23 and 17.24 together are known the Rayleigh-Ritz theorem. | No |
Proposition 17.25. Let \( A \) be an \( n \times n \) symmetric matrix, \( R \) be an \( n \times m \) matrix such that \( {R}^{\top }R = I \) (with \( m \leq n \) ), and let \( B = {R}^{\top }{AR} \) (an \( m \times m \) matrix). The following properties hold:\n\n(a) The eigenvalues of \( B \) interlace the eigenvalue... | Proof. (a) Let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be an orthonormal basis of eigenvectors for \( A \), and let \( \left( {{v}_{1},\ldots ,{v}_{m}}\right) \) be an orthonormal basis of eigenvectors for \( B \) . Let \( {U}_{j} \) be the subspace spanned by \( \left( {{u}_{1},\ldots ,{u}_{j}}\right) \) and let... | Yes |
Theorem 17.27. (Courant-Fischer) Let \( A \) be a symmetric \( n \times n \) matrix with eigenvalues \( {\lambda }_{1} \leq {\lambda }_{2} \leq \cdots \leq {\lambda }_{n} \) . If \( {\mathcal{V}}_{k} \) denotes the set of subspaces of \( {\mathbb{R}}^{n} \) of dimension \( k \), then\n\n\[ \n{\lambda }_{k} = \mathop{\m... | Proof. Let us consider the second equality, the proof of the first equality being similar. Let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be any orthonormal basis of eigenvectors of \( A \), where \( {u}_{i} \) is a unit eigenvector associated with \( {\lambda }_{i} \) . Observe that the space \( {V}_{k} \) spanned ... | Yes |
Proposition 17.28. Given two \( n \times n \) symmetric matrices \( A \) and \( B = A + {\Delta A} \), if \( {\alpha }_{1} \leq {\alpha }_{2} \leq \cdots \leq {\alpha }_{n} \) are the eigenvalues of \( A \) and \( {\beta }_{1} \leq {\beta }_{2} \leq \cdots \leq {\beta }_{n} \) are the eigenvalues of \( B \), then\n\n\[... | Proof. Let \( {\mathcal{V}}_{k} \) be defined as in the Courant-Fischer theorem and let \( {V}_{k} \) be the subspace spanned by the \( k \) eigenvectors associated with \( {\lambda }_{1},\ldots ,{\lambda }_{k} \). By the Courant-Fischer theorem applied to \( B \), we have\n\n\[ {\beta }_{k} = \mathop{\min }\limits_{{W... | Yes |
Proposition 17.29. (Weyl) Given two symmetric (or Hermitian) \( n \times n \) matrices \( A \) and \( B \) , the following inequalities hold: For all \( i, j, k \) with \( 1 \leq i, j, k \leq n \) :\n\n1. If \( i + j = k + 1 \), then\n\n\[{\lambda }_{i}\left( A\right) + {\lambda }_{j}\left( B\right) \leq {\lambda }_{k}... | Proof. Observe that the first set of inequalities is obtained form the second set by replacing \( A \) by \( - A \) and \( B \) by \( - B \), so it is enough to prove the second set of inequalities. By the Courant-Fischer theorem, there is a subspace \( H \) of dimension \( n - k + 1 \) such that\n\n\[{\lambda }_{k}\le... | Yes |
If \( A = \left( \begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \end{array}\right) \), running Hessenberg1 we find \( H \) and \( Q \). | \[ H = \left( \begin{matrix} {1.0000} & - {5.3852} & 0 & 0 \\ - {5.3852} & {15.2069} & - {1.6893} & - {0.0000} \\ - {0.0000} & - {1.6893} & - {0.2069} & - {0.0000} \\ 0 & - {0.0000} & {0.0000} & {0.0000} \end{matrix}\right) \] \[ Q = \left( \begin{matrix} {1.0000} & 0 & 0 & 0 \\ 0 & - {0.3714} & - {0.5571} & - {0.7428}... | Yes |
Theorem 18.4. Let \( H \) be a symmetric (or Hermitian) positive definite tridiagonal matrix. If \( H \) is unreduced, then the QR algorithm converges to a diagonal matrix consisting of the eigenvalues of \( H \) . | Since every symmetric (or Hermitian) positive definite matrix is similar to tridiagonal symmetric (resp. Hermitian) positive definite matrix, we deduce that we have a method for finding the eigenvalues of a symmetric (resp. Hermitian) positive definite matrix (more accurately, to find approximations as good as we want ... | Yes |
Consider a beam of unit length supported at its ends in 0 and 1, stretched along its axis by a force \( P \), and subjected to a transverse load \( f\left( x\right) {dx} \) per element \( {dx} \), as illustrated in Figure 19.1. The bending moment \( u\left( x\right) \) at the absissa \( x \) is the solution of a bounda... | If we seek a solution \( u \in {C}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \), that is, a function whose first and second derivatives exist and are continuous, then it can be shown that the problem has a unique solution (assuming \( c \) and \( f \) to be continuous functions on \( \left\lbrack {0,1}\right\rb... | Yes |
Theorem 19.1. Let \( u \) be any solution of the boundary problem (BP).\n\n(1) Then we have\n\n\[ a\left( {u, v}\right) = \widetilde{f}\left( v\right) ,\;\text{ for all }v \in V, \]\n\n(WF)\n\nwhere\n\n\[ a\left( {u, v}\right) = {\int }_{0}^{1}\left( {{u}^{\prime }{v}^{\prime } + {cuv}}\right) {dx},\;\text{ for all }u,... | Proof. We already proved (1).\n\nTo prove (2), first we show that\n\n\[ \parallel v{\parallel }_{V}^{2} \leq {2a}\left( {v, v}\right) ,\;\text{ for all }v \in V. \]\n\nFor this, it suffices to prove that\n\n\[ \parallel v{\parallel }_{V}^{2} \leq 2{\int }_{0}^{1}{\left( {f}^{\prime }\left( x\right) \right) }^{2}{dx},\;... | Yes |
Theorem 19.2. Suppose \( c\left( x\right) \geq 0 \) for all \( x \in \left\lbrack {0,1}\right\rbrack \) . For every finite-dimensional subspace \( {V}_{a}\left( {\dim \left( {V}_{a}\right) = n}\right) \) of \( V \), for every basis \( \left( {{w}_{1},\ldots ,{w}_{n}}\right) \) of \( {V}_{a} \), the following properties... | We proved (1) and (2), but we will omit the proof of (3) which can be found in Ciarlet [41]. | No |
Proposition 20.1. Let \( G = \left( {V, E}\right) \) be any undirected graph with \( m \) vertices, \( n \) edges, and \( c \) connected components. For any orientation \( \sigma \) of \( G \), if \( B \) is the incidence matrix of the oriented graph \( {G}^{\sigma } \), then \( c = \dim \left( {\operatorname{Ker}\left... | Proof. (After Godsil and Royle [77], Section 8.3). The fact that \( \operatorname{rank}\left( B\right) = m - c \) will be proved last.\n\nLet us prove that the kernel of \( {B}^{\top } \) has dimension \( c \) . A vector \( z \in {\mathbb{R}}^{m} \) belongs to the kernel of \( {B}^{\top } \) iff \( {B}^{\top }z = 0 \) ... | Yes |
Proposition 20.2. Given any undirected graph \( G \), for any orientation \( \sigma \) of \( G \), if \( {B}^{\sigma } \) is the incidence matrix of the directed graph \( {G}^{\sigma }, A \) is the adjacency matrix of \( {G}^{\sigma } \), and \( D \) is the degree matrix such that \( {D}_{ii} = d\left( {v}_{i}\right) \... | Proof. The entry \( {B}^{\sigma }{\left( {B}^{\sigma }\right) }_{ij}^{\top } \) is the inner product of the \( i \) th row \( {b}_{i}^{\sigma } \), and the \( j \) th row \( {b}_{j}^{\sigma } \) of \( {B}^{\sigma } \) . If \( i = j \), then as\n\n\[ \n{b}_{ik}^{\sigma } = \left\{ \begin{array}{ll} + 1 & \text{ if }s\le... | Yes |
Proposition 20.3. Given any weighted graph \( G = \left( {V, W}\right) \) with \( V = \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \), if \( {B}^{\sigma } \) is the incidence matrix of any oriented graph \( {G}^{\sigma } \) obtained from the underlying graph of \( G \) and \( D \) is the degree matrix of \( G \), then\n\n... | Consequently, \( {B}^{\sigma }{\left( {B}^{\sigma }\right) }^{\top } \) is independent of the orientation of the underlying graph of \( G \) and \( L = D - W \) is symmetric and positive semidefinite; that is, the eigenvalues of \( L = D - W \) are real and nonnegative. | Yes |
Proposition 20.4. For any \( m \times m \) symmetric matrix \( W = \left( {w}_{ij}\right) \), if we let \( L = D - W \) where \( D \) is the degree matrix associated with \( W \) (that is, \( {d}_{i} = \mathop{\sum }\limits_{{j = 1}}^{m}{w}_{ij} \) ), then we have\n\n\[ \n{x}^{\top }{Lx} = \frac{1}{2}\mathop{\sum }\lim... | Proof. We have\n\n\[ \n{x}^{\top }{Lx} = {x}^{\top }{Dx} - {x}^{\top }{Wx}\n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1}}^{m}{d}_{i}{x}_{i}^{2} - \mathop{\sum }\limits_{{i, j = 1}}^{m}{w}_{ij}{x}_{i}{x}_{j}\n\]\n\n\[ \n= \frac{1}{2}\left( {\mathop{\sum }\limits_{{i = 1}}^{m}{d}_{i}{x}_{i}^{2} - 2\mathop{\sum }\limits_{{... | Yes |
As an example, the matrices \( {L}_{\mathrm{{sym}}} \) and \( {L}_{\mathrm{{rw}}} \) associated with the graph \( {G}_{1} \) are\n\n\[ \n{L}_{\mathrm{{sym}}} = \left( \begin{matrix} {1.0000} & - {0.3536} & - {0.4082} & 0 & 0 \\ - {0.3536} & {1.0000} & - {0.2887} & - {0.2887} & - {0.3536} \\ - {0.4082} & - {0.2887} & {1... | Since the unnormalized Laplacian \( L \) can be written as \( L = B{B}^{\top } \), where \( B \) is the incidence matrix of any oriented graph obtained from the underlying graph of \( G = \left( {V, W}\right) \), if we let \n\n\[ \n{B}_{\text{sym }} = {D}^{-1/2}B \n\] \n\nwe get \n\n\[ \n{L}_{\mathrm{{sym}}} = {B}_{\ma... | Yes |
(1) The matrix \( {L}_{\text{sym }} \) is symmetric and positive semidefinite. In fact,\n\n\[ \n{x}^{\top }{L}_{\mathrm{{sym}}}x = \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{m}{w}_{ij}{\left( \frac{{x}_{i}}{\sqrt{{d}_{i}}} - \frac{{x}_{j}}{\sqrt{{d}_{j}}}\right) }^{2}\;\text{ for all }x \in {\mathbb{R}}^{m}.\n\] | Proof. (1) We have \( {L}_{\mathrm{{sym}}} = {D}^{-1/2}L{D}^{-1/2} \), and \( {D}^{-1/2} \) is a symmetric invertible matrix (since it is an invertible diagonal matrix). It is a well-known fact of linear algebra that if \( B \) is an invertible matrix, then a matrix \( S \) is symmetric, positive semidefinite iff \( {B... | Yes |
Prove that if \( x = {D}^{-1/2}y \), then \( R = \frac{{x}^{\top }{Lx}}{{\left( {D}^{1/2}x\right) }^{\top }{D}^{1/2}x} = \frac{\mathop{\sum }\limits_{{u \sim v}}{\left( x\left( u\right) - x\left( v\right) \right) }^{2}}{\mathop{\sum }\limits_{v}{d}_{v}x{\left( v\right) }^{2}} \). | \[ R = \frac{{y}^{\top }{L}_{\text{sym }}y}{{y}^{\top }y}. \] | No |
Proposition 21.1. Let \( G = \left( {V, W}\right) \) be a weighted graph, with \( \left| V\right| = m \) and \( W \) an \( m \times m \) symmetric matrix, and let \( R \) be the matrix of a graph drawing \( \rho \) of \( G \) in \( {\mathbb{R}}^{n} \) (a \( m \times n \) matrix). If \( L = D - W \) is the unnormalized ... | Proof. Since \( \rho \left( {v}_{i}\right) \) is the \( i \) th row of \( R \) (and \( \rho \left( {v}_{j}\right) \) is the \( j \) th row of \( R \) ), if we denote the \( k \) th column of \( R \) by \( {R}^{k} \), using Proposition 20.4, we have\n\n\[ \mathcal{E}\left( R\right) = \mathop{\sum }\limits_{{\left\{ {{v}... | Yes |
Theorem 21.2. Let \( G = \left( {V, W}\right) \) be a weighted graph with \( \left| V\right| = m \) . If \( L = D - W \) is the (unnormalized) Laplacian of \( G \), and if the eigenvalues of \( L \) are \( 0 = {\lambda }_{1} < {\lambda }_{2} \leq {\lambda }_{3} \leq \ldots \leq {\lambda }_{m} \) , then the minimal ener... | Proof. We present the proof given in Godsil and Royle [77] (Section 13.4, Theorem 13.4.1). The key point is that the sum of the \( n \) smallest eigenvalues of \( L \) is a lower bound for \( \operatorname{tr}\left( {{R}^{\top }{LR}}\right) \) . This can be shown using a Rayleigh ratio argument; see Proposition 17.25 (... | Yes |
Consider the graph with four nodes whose adjacency matrix is\n\n\[ A = \left( \begin{array}{llll} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array}\right) \] | We use the following program to compute \( {u}_{2} \) and \( {u}_{3} \) :\n\n\( A = \left\lbrack {{0110};{1001};{1001};{0110}}\right\rbrack ; \)\n\n\( \mathrm{D} = \operatorname{diag}\left( {\operatorname{sum}\left( \mathrm{A}\right) }\right) \) ;\n\n\( \mathrm{L} = \mathrm{D} - \mathrm{A}; \)\n\n\( \left\lbrack {v, e}... | Yes |
Consider the graph \( {G}_{2} \) shown in Figure 20.3 given by the adjacency matrix\n\n\[ A = \left( \begin{array}{lllll} 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \end{array}\right) \] | We use the following program to compute \( {u}_{2} \) and \( {u}_{3} \) :\n\n\( \mathbf{A} = \left\lbrack {{01}\mathbf{1}0\mathbf{0};\mathbf{1}0\mathbf{1}\mathbf{1}\mathbf{1};\mathbf{1}\mathbf{1}0\mathbf{1}0;0\mathbf{1}\mathbf{1}0\mathbf{1};0\mathbf{1}0\mathbf{1}0}\right\rbrack ; \)\n\n\( \mathrm{D} = \operatorname{dia... | No |
Consider the ring graph defined by the adjacency matrix \( A \) given in the Matlab program shown below: | \n\nFigure 21.2: Drawing of the graph from Example 2.\n\n\( A = \operatorname{diag}\left( {\operatorname{ones}\left( {1,{11}}\right) ,1}\right) \) ;\n\nA = A + A ’;\n\n\( A\left( {1,{12}}\right) = 1;A\left( {{12},1}\... | Yes |
In this example adapted from Spielman, we generate 20 randomly chosen points in the unit square, compute their Delaunay triangulation, then the adjacency matrix of the corresponding graph, and finally draw the graph using the second and third eigenvalues of the Laplacian. | \n\( \\mathrm{A} = \\operatorname{zeros}\\left( {{20},{20}}\\right) \) ;\n\n\( \\mathrm{{xy}} = \\operatorname{rand}\\left( {{20},2}\\right) \) ;\n\ntrigs \( = \\operatorname{delaunay}\\left( {\\mathrm{{xy}}\\left( { : ,1}\\right) ,\\mathrm{{xy}}\\left( { : ,2}\\right) }\\right) \) ;\n\nelemtrig \( = \\operatorname{one... | Yes |
What we should really do is to plot this graph in \( {\mathbb{R}}^{3} \) using three orthonormal eigenvectors associated with \( {\lambda }_{2} \) . | A 3D picture of the graph of the Buckyball is produced by the following Matlab program, and its image is shown in Figure 21.6. It looks better!\n\n\( \left\lbrack {x, y}\right\rbrack = \operatorname{gplot}\left( {A, v\left( { : ,\left\lbrack {23}\right\rbrack }\right) }\right) \) ;\n\n[x, z] = gplot(A, v(:, [2 4]));\n\... | Yes |
Proposition 22.1. The eigenvalues of \( {f}^{ * } \circ f \) and \( f \circ {f}^{ * } \) are nonnegative. | Proof. If \( u \) is an eigenvector of \( {f}^{ * } \circ f \) for the eigenvalue \( \lambda \), then\n\n\[ \left\langle {\left( {{f}^{ * } \circ f}\right) \left( u\right), u}\right\rangle = \langle f\left( u\right), f\left( u\right) \rangle \]\n\nand\n\n\[ \left\langle {\left( {{f}^{ * } \circ f}\right) \left( u\right... | Yes |
Theorem 22.3. (Singular value decomposition) For every real \( n \times n \) matrix \( A \) there are two orthogonal matrices \( U \) and \( V \) and a diagonal matrix \( D \) such that \( A = {VD}{U}^{\top } \), where \( D \) is of the form\n\n\[ D = \left( \begin{matrix} {\sigma }_{1} & & \ldots & \\ & {\sigma }_{2} ... | Proof. Since \( {A}^{\top }A \) is a symmetric matrix, in fact, a positive semidefinite matrix, there exists an orthogonal matrix \( U \) such that\n\n\[ {A}^{\top }A = U{D}^{2}{U}^{\top } \]\n\nwith \( D = \operatorname{diag}\left( {{\sigma }_{1},\ldots ,{\sigma }_{r},0,\ldots ,0}\right) \), where \( {\sigma }_{1}^{2}... | Yes |
Here is a simple example of how to use the proof of Theorem 22.3 to obtain an SVD decomposition. Let \( A = \left( \begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right) \) . | Then \( {A}^{\top } = \left( \begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right) ,{A}^{\top }A = \left( \begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \), and \( A{A}^{\top } = \left( \begin{array}{ll} 2 & 0 \\ 0 & 0 \end{array}\right) \) . A simple calculation shows that the eigenvalues of \( {A}^{\top }A \) are ... | No |
Recall from Example 22.1 that \( A = {VD}{U}^{\top } \) where \( V = {I}_{2} \) and | \[ A = \left( \begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right) ,\;U = \left( \begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}\right) ,\;D = \left( \begin{matrix} \sqrt{2} & 0 \\ 0 & 0 \end{matrix}\right) . \] Set \( R = V{U}^{\top } = U \) and \[ S = {U... | No |
Let \( A = \left( \begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right) \) and \( A = {R}_{1}S \), where \( {R}_{1} = \left( \begin{matrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & - 1/\sqrt{2} \end{matrix}\right) \) and \( S = \left( \begin{array}{ll} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{array}\right) ... | \[ S = \left( \begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}\right) \left( \begin{matrix} \sqrt{2} & 0 \\ 0 & 0 \end{matrix}\right) \left( \begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}\ri... | Yes |
Theorem 22.4. (Weyl’s inequalities,1949) For any complex \( n \times n \) matrix, \( A \), if \( {\lambda }_{1},\ldots ,{\lambda }_{n} \in \) \( \mathbb{C} \) are the eigenvalues of \( A \) and \( {\sigma }_{1},\ldots ,{\sigma }_{n} \in {\mathbb{R}}_{ + } \) are the singular values of \( A \), listed so that \( \left| ... | A proof of Theorem 22.4 can be found in Horn and Johnson [94], Chapter 3, Section 3.3, where more inequalities relating the eigenvalues and the singular values of a matrix are given. | No |
Theorem 22.5. (Singular value decomposition) For every real \( m \times n \) matrix \( A \), there are two orthogonal matrices \( U\left( {n \times n}\right) \) and \( V\left( {m \times m}\right) \) and a diagonal \( m \times n \) matrix \( D \) such that \( A = {VD}{U}^{\top } \), where \( D \) is of the form\n\n\[ D ... | Proof. As in the proof of Theorem 22.3, since \( {A}^{\top }A \) is symmetric positive semidefinite, there exists an \( n \times n \) orthogonal matrix \( U \) such that\n\n\[ {A}^{\top }A = U{\sum }^{2}{U}^{\top } \]\n\nwith \( \sum = \operatorname{diag}\left( {{\sigma }_{1},\ldots ,{\sigma }_{r},0,\ldots ,0}\right) \... | Yes |
Problem 22.1. (1) Let \( A \) be a real \( n \times n \) matrix and consider the \( \left( {2n}\right) \times \left( {2n}\right) \) real symmetric matrix \[ S = \left( \begin{matrix} 0 & A \\ {A}^{\top } & 0 \end{matrix}\right) \] Suppose that \( A \) has rank \( r \) . If \( A = {V\sum }{U}^{\top } \) is an SVD for \(... | Hint. We have \( A{u}_{k} = {\sigma }_{k}{v}_{k} \) for \( k = 1,\ldots, n \) . Show that \( {A}^{\top }{v}_{k} = {\sigma }_{k}{u}_{k} \) for \( k = 1,\ldots, r \), and that \( {A}^{\top }{v}_{k} = 0 \) for \( k = r + 1,\ldots, n \) . Recall that \( \operatorname{Ker}\left( {A}^{\top }\right) = \operatorname{Ker}\left(... | No |
Problem 22.2. Let \( A \) be a real \( m \times n \) matrix of rank \( r \) and let \( q = \min \left( {m, n}\right) \). Consider the \( \left( {m + n}\right) \times \left( {m + n}\right) \) real symmetric matrix \[ S = \left( \begin{matrix} 0 & A \\ {A}^{\top } & 0 \end{matrix}\right) \] and prove that \[ \left( \begi... | \[ \left( \begin{matrix} {I}_{m} & {z}^{-1}A \\ 0 & {I}_{n} \end{matrix}\right) \left( \begin{matrix} z{I}_{m} & - A \\ - {A}^{\top } & z{I}_{n} \end{matrix}\right) = \left( \begin{matrix} z{I}_{m} - {z}^{-1}A{A}^{\top } & 0 \\ - {A}^{\top } & z{I}_{n} \end{matrix}\right) . \] | Yes |
Problem 22.9. Let \( A \) be a real \( n \times n \) matrix.\n\n(1) Assume \( A \) is invertible. Prove that if \( A = {Q}_{1}{S}_{1} = {Q}_{2}{S}_{2} \) are two polar decompositions of \( A \), then \( {Q}_{1} = {Q}_{2} \) and \( {S}_{1} = {S}_{2} \). | Hint. \( {A}^{\top }A = {S}_{1}^{2} = {S}_{2}^{2} \), with \( {S}_{1} \) and \( {S}_{2} \) symmetric positive definite. Then use Problem 17.7. | No |
As a concrete illustration, suppose that we observe the motion of a small object, assimilated to a point, in the plane. From our observations, we suspect that this point moves along a straight line, say of equation \( y = {dx} + c \) . Suppose that we observed the moving point at three different locations \( \left( {{x... | The idea of the method of least squares is to determine \( \left( {c, d}\right) \) such that it minimizes the sum of the squares of the errors, namely,\n\n\[ {\left( c + d{x}_{1} - {y}_{1}\right) }^{2} + {\left( c + d{x}_{2} - {y}_{2}\right) }^{2} + {\left( c + d{x}_{3} - {y}_{3}\right) }^{2}. \] | Yes |
The least squares solution of smallest norm of the linear system \( {Ax} = b \) , where \( A \) is an \( m \times n \) matrix, is given by | \[ {x}^{ + } = {A}^{ + }b = U{D}^{ + }{V}^{\top }b. \] Proof. First assume that \( A \) is a (rectangular) diagonal matrix \( D \), as above. Then since \( x \) minimizes \( \parallel {Dx} - b{\parallel }_{2}^{2} \) iff \( {Dx} \) is the projection of \( b \) onto the image subspace \( F \) of \( D \), it is fairly obv... | Yes |
Proposition 23.3. When \( A \) has full rank, the pseudo-inverse \( {A}^{ + } \) can be expressed as \( {A}^{ + } = \) \( {\left( {A}^{\top }A\right) }^{-1}{A}^{\top } \) when \( m \geq n \), and as \( {A}^{ + } = {A}^{\top }{\left( A{A}^{\top }\right) }^{-1} \) when \( n \geq m \) . In the first case \( \left( {m \geq... | Proof. If \( m \geq n \) and \( A \) has full rank \( n \), we have\n\n\[ A = V\left( \begin{matrix} \Lambda \\ {0}_{m - n, n} \end{matrix}\right) {U}^{\top } \]\n\nwith \( \Lambda \) an \( n \times n \) diagonal invertible matrix (with positive entries), so\n\n\[ {A}^{ + } = U\left( \begin{array}{ll} {\Lambda }^{-1} &... | Yes |
Proposition 23.4. The matrix \( A{A}^{ + } \) is the orthogonal projection onto the range of \( A \) and \( {A}^{ + }A \) is the orthogonal projection onto \( \operatorname{Ker}{\left( A\right) }^{ \bot } = \operatorname{Im}\left( {A}^{\top }\right) \), the range of \( {A}^{\top } \) . | Proof. Obviously, we have \( \operatorname{range}\left( {A{A}^{ + }}\right) \subseteq \operatorname{range}\left( A\right) \), and for any \( y = {Ax} \in \operatorname{range}\left( A\right) \), since \( A{A}^{ + }A = A \), we have\n\n\[ A{A}^{ + }y = A{A}^{ + }{Ax} = {Ax} = y, \]\n\nso the image of \( A{A}^{ + } \) is ... | Yes |
Proposition 23.5. The set range \( \left( A\right) = \operatorname{range}\left( {A{A}^{ + }}\right) \) consists of all vectors \( y \in {\mathbb{R}}^{m} \) such that\n\n\[ \n{V}^{\top }y = \left( \begin{array}{l} z \\ 0 \end{array}\right)\n\]\n\nwith \( z \in {\mathbb{R}}^{r} \) . | Proof. Indeed, if \( y = {Ax} \), then\n\n\[ \n{V}^{\top }y = {V}^{\top }{Ax} = {V}^{\top }{V\sum }{U}^{\top }x = \sum {U}^{\top }x = \left( \begin{matrix} {\sum }_{r} & 0 \\ 0 & {0}_{m - r} \end{matrix}\right) {U}^{\top }x = \left( \begin{array}{l} z \\ 0 \end{array}\right) ,\n\]\n\nwhere \( {\sum }_{r} \) is the \( r... | Yes |
Proposition 23.6. The set range \( \left( {{A}^{ + }A}\right) = \operatorname{Ker}{\left( A\right) }^{ \bot } \) consists of all vectors \( y \in {\mathbb{R}}^{n} \) such that\n\n\[ \n{U}^{\top }y = \left( \begin{array}{l} z \\ 0 \end{array}\right)\n\]\nwith \( z \in {\mathbb{R}}^{r} \) . | Proof. If \( y = {A}^{ + }{Au} \), then\n\n\[ \ny = {A}^{ + }{Au} = U\left( \begin{matrix} {I}_{r} & 0 \\ 0 & {0}_{n - r} \end{matrix}\right) {U}^{\top }u = U\left( \begin{array}{l} z \\ 0 \end{array}\right) ,\n\]\n\nfor some \( z \in {\mathbb{R}}^{r} \) . Conversely, if \( {U}^{\top }y = \left( \begin{array}{l} z \\ 0... | Yes |
For any (real) normal matrix \( A \) and any block diagonalization \( A = \) \( {U\Lambda }{U}^{\top } \) of \( A \) as above, the pseudo-inverse of \( A \) is given by\n\n\[ \n{A}^{ + } = U{\Lambda }^{ + }{U}^{\top }\n\]\n\nwhere \( {\Lambda }^{ + } \) is the pseudo-inverse of \( \Lambda \) . Furthermore, if\n\n\[ \n\... | Proof. Assume that \( {B}_{1},\ldots ,{B}_{p} \) are \( 2 \times 2 \) blocks and that \( {\lambda }_{{2p} + 1},\ldots ,{\lambda }_{n} \) are the scalar entries. We know that the numbers \( {\lambda }_{j} \pm i{\mu }_{j} \), and the \( {\lambda }_{{2p} + k} \) are the eigenvalues of \( A \) . Let \( {\rho }_{{2j} - 1} =... | Yes |
Consider the following real diagonal form of the normal matrix\n\n\[ A = \left( \begin{matrix} - {2.7500} & {2.1651} & - {0.8660} & {0.5000} \\ {2.1651} & - {0.2500} & - {1.5000} & {0.8660} \\ {0.8660} & {1.5000} & {0.7500} & - {0.4330} \\ - {0.5000} & - {0.8660} & - {0.4330} & {0.2500} \end{matrix}\right) = {U\Lambda ... | with\n\n\[ U = \left( \begin{matrix} \cos \left( {\pi /3}\right) & 0 & \sin \left( {\pi /3}\right) & 0 \\ \sin \left( {\pi /3}\right) & 0 & - \cos \left( {\pi /3}\right) & 0 \\ 0 & \cos \left( {\pi /6}\right) & 0 & \sin \left( {\pi /6}\right) \\ 0 & - \cos \left( {\pi /6}\right) & 0 & \sin \left( {\pi /6}\right) \end{m... | Yes |
Proposition 23.9. Let \( A \) be an \( m \times n \) matrix of rank \( r \) and let \( {VD}{U}^{\top } = A \) be an SVD for A. Write \( {u}_{i} \) for the columns of \( U,{v}_{i} \) for the columns of \( V \), and \( {\sigma }_{1} \geq {\sigma }_{2} \geq \cdots \geq {\sigma }_{p} \) for the singular values of \( A\left... | Proof. By construction, \( {A}_{k} \) has rank \( k \), and we have\n\n\[ \n{\begin{Vmatrix}A - {A}_{k}\end{Vmatrix}}_{2} = {\begin{Vmatrix}\mathop{\sum }\limits_{{i = k + 1}}^{p}{\sigma }_{i}{v}_{i}{u}_{i}^{\top }\end{Vmatrix}}_{2} = {\begin{Vmatrix}V\operatorname{diag}\left( 0,\ldots ,0,{\sigma }_{k + 1},\ldots ,{\si... | Yes |
Consider the badly conditioned symmetric matrix\n\n\[ A = \left( \begin{matrix} {10} & 7 & 8 & 7 \\ 7 & 5 & 6 & 5 \\ 8 & 6 & {10} & 9 \\ 7 & 5 & 9 & {10} \end{matrix}\right) \] | Since \( A \) is SPD, we have the SVD\n\n\[ A = {UD}{U}^{\top }, \]\nwith\n\n\[ U = \left( \begin{matrix} - {0.5286} & - {0.6149} & {0.3017} & - {0.5016} \\ - {0.3803} & - {0.3963} & - {0.0933} & {0.8304} \\ - {0.5520} & {0.2716} & - {0.7603} & - {0.2086} \\ - {0.5209} & {0.6254} & {0.5676} & {0.1237} \end{matrix}\righ... | Yes |
If we take \( x = \\left( {1,3, - 1}\\right) \) and \( y = \\left( {0,2, - 2}\\right) \), we know from Example 23.6 that \( x - \\bar{x} = \\left( {0,2, - 2}\\right) \) and \( y - \\bar{y} = \\left( {-1,0,1}\\right) \). Thus, \( \\operatorname{cov}\\left( {x, y}\\right) = \\frac{0\\left( {-1}\\right) + 2\\left( 0\\righ... | The covariance of \( x \) and \( y \) measures how \( x \) and \( y \) vary from the mean with respect to each other. Obviously, \( \\operatorname{cov}\\left( {x, y}\\right) = \\operatorname{cov}\\left( {y, x}\\right) \) and \( \\operatorname{cov}\\left( {x, x}\\right) = \\operatorname{var}\\left( x\\right) \). Note th... | Yes |
Let \( X = \left( \begin{matrix} 1 & 1 \\ 3 & 2 \\ - 1 & 3 \end{matrix}\right) \), the \( 3 \times 2 \) matrix whose columns are the vector \( x \) and \( y \) of Example 23.6. Then | \[ \mu = \frac{1}{3}\left\lbrack {\left( {1,1}\right) + \left( {3,2}\right) + \left( {-1,3}\right) }\right\rbrack = \left( {1,2}\right) , \] \[ X - \mu = \left( \begin{matrix} 0 & - 1 \\ 2 & 0 \\ - 2 & 1 \end{matrix}\right) \] and \[ \sum = \frac{1}{2}\left( \begin{matrix} 0 & 2 & - 2 \\ - 1 & 0 & 1 \end{matrix}\right)... | Yes |
If \( v \in {\mathbb{R}}^{d} \) is a unit vector, we wish to consider the projection of the data points \( {X}_{1},\ldots ,{X}_{n} \) onto the line spanned by \( v \) . Recall from Euclidean geometry that if \( x \in {\mathbb{R}}^{d} \) is any vector and \( v \in {\mathbb{R}}^{d} \) is a unit vector, the projection of ... | \[ \langle x, v\rangle v\text{.} \] Thus, with respect to the basis \( v \), the projection of \( x \) has coordinate \( \langle x, v\rangle \) . If \( x \) is represented by a row vector and \( v \) by a column vector, then \[ \langle x, v\rangle = {xv}. \] Therefore, the vector \( Y \in {\mathbb{R}}^{n} \) consisting... | Yes |
Proposition 23.10. If \( A \) is a symmetric \( d \times d \) matrix with eigenvalues \( {\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \geq \) \( {\lambda }_{d} \) and if \( \left( {{u}_{1},\ldots ,{u}_{d}}\right) \) is any orthonormal basis of eigenvectors of \( A \), where \( {u}_{i} \) is a unit eigenvector associa... | Proof. First observe that\n\n\[ \mathop{\max }\limits_{{x \neq 0}}\frac{{x}^{\top }{Ax}}{{x}^{\top }x} = \mathop{\max }\limits_{x}\left\{ {{x}^{\top }{Ax} \mid {x}^{\top }x = 1}\right\} \]\nand similarly,\n\n\[ \mathop{\max }\limits_{{x \neq 0, x \in {\left\{ {u}_{1},\ldots ,{u}_{k}\right\} }^{ \bot }}}\frac{{x}^{\top ... | Yes |
For the centered data set of our bearded mathematicians (Example 23.9) we have \( X - \mu = {V\sum }{U}^{\top } \), where \( \sum \) has two nonzero singular values, \( {\sigma }_{1} = {116.9803},{\sigma }_{2} = \) 21.7812, and with\n\n\[ U = \left( \begin{matrix} {0.9995} & {0.0325} \\ {0.0325} & - {0.9995} \end{matri... | See Figures 23.4, 23.5, and 23.6. | No |
Theorem 23.12. Let \( X \) be an \( n \times d \) matrix of data points \( {X}_{1},\ldots ,{X}_{n} \), and let \( \mu \) be the centroid of the \( {X}_{i} \) ’s. If \( X - \mu = {VD}{U}^{\top } \) is an SVD decomposition of \( X - \mu \) and if the main diagonal of \( D \) consists of the singular values \( {\sigma }_{... | \[ {A}_{k} = \mu + {U}_{d - k} \] where \( {U}_{d - k} \) is the linear subspace spanned by the first \( d - k \) columns of \( U \), the first \( d - k \) principal directions of \( X - \mu \left( {1 \leq k \leq d - 1}\right) \) . | Yes |
Suppose in the data set of Example 23.5 that we add the month of birth of every mathematician as a feature. We obtain the following data set. | The mean of the first column is 5.2 , and the centered data set is given below. Running SVD on this data set we get \[ U = \left( \begin{matrix} {0.0394} & {0.1717} & {0.9844} \\ - {0.9987} & {0.0390} & {0.0332} \\ - {0.0327} & - {0.9844} & {0.1730} \end{matrix}\right) \] \[ D = \left( \begin{matrix} {117.0706} & 0 & 0... | Yes |
Consider the overdetermined system in the single variable \( x \) :\n\n\[ \n{a}_{1}x = {b}_{1},\ldots ,{a}_{m}x = {b}_{m}.\n\] | Prove that the least squares solution of smallest norm is given by\n\n\[ \n{x}^{ + } = \frac{{a}_{1}{b}_{1} + \cdots + {a}_{m}{b}_{m}}{{a}_{1}^{2} + \cdots + {a}_{m}^{2}}.\n\] | No |
Prove that\n\n\[ \parallel A - B{\parallel }_{F}^{2} = \parallel A - C{\parallel }_{F}^{2} + \parallel L{\parallel }_{F}^{2} + \parallel R{\parallel }_{F}^{2} + \parallel F{\parallel }_{F}^{2}. \] | Since \( \parallel A - B{\parallel }_{F} \) is minimal, show that \( L = R = F = 0 \. | No |
Proposition 24.1. Given an affine space \( E \), let \( {\left( {a}_{i}\right) }_{i \in I} \) be a family of points in \( E \), and let \( {\left( {\lambda }_{i}\right) }_{i \in I} \) be a family of scalars. For any two points \( a, b \in E \), the following properties hold:\n\n(1) If \( \mathop{\sum }\limits_{{i \in I... | Proof. (1) By Chasles's identity (see Section 24.3), we have\n\n\[ a + \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i}\overrightarrow{a{a}_{i}} = a + \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i}\left( {\overrightarrow{ab} + \overrightarrow{b{a}_{i}}}\right) \]\n\n\[ = a + \left( {\mathop{\sum }\limits_{{i \in I}}{\l... | Yes |
Proposition 24.2. Let \( \langle E,\overrightarrow{E}, + \rangle \) be an affine space.\n\n(1) A nonempty subset \( V \) of \( E \) is an affine subspace iff for every point \( a \in V \), the set\n\n\[ \n{\overrightarrow{V}}_{a} = \{ \overrightarrow{ax} \mid x \in V\}\n\]\n\nis a subspace of \( \overrightarrow{E} \) .... | Proof. The proof is straightforward, and is omitted. It is also given in Gallier [71]. | No |
Proposition 24.3. Given an affine space \( \langle E,\overrightarrow{E}, + \rangle \), for any family \( {\left( {a}_{i}\right) }_{i \in I} \) of points in \( E \), the set \( V \) of barycenters \( \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i}{a}_{i} \) (where \( \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i} = 1 \... | Proof. If \( {\left( {a}_{i}\right) }_{i \in I} \) is empty, then \( V = \varnothing \), because of the condition \( \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i} = 1 \) . If \( {\left( {a}_{i}\right) }_{i \in I} \) is nonempty, then the smallest affine subspace containing \( {\left( {a}_{i}\right) }_{i \in I} \) mus... | Yes |
Proposition 24.4. Given an affine space \( \langle E,\overrightarrow{E}, + \rangle \), let \( {\left( {a}_{i}\right) }_{i \in I} \) be a family of points in \( E \) . If the family \( {\left( \overrightarrow{{a}_{i}{a}_{j}}\right) }_{j \in \left( {I-\{ i\} }\right) } \) is linearly independent for some \( i \in I \), t... | Proof. Assume that the family \( {\left( \overrightarrow{{a}_{i}{a}_{j}}\right) }_{j \in \left( {I-\{ i\} }\right) } \) is linearly independent for some specific \( i \in I \) . Let \( k \in I \) with \( k \neq i \), and assume that there are some scalars \( {\left( {\lambda }_{j}\right) }_{j \in \left( {I-\{ k\} }\rig... | Yes |
Proposition 24.5. Given an affine space \( \langle E,\overrightarrow{E}, + \rangle \), let \( \left( {{a}_{0},\ldots ,{a}_{m}}\right) \) be a family of \( m + 1 \) points in \( E \). Let \( x \in E \), and assume that \( x = \mathop{\sum }\limits_{{i = 0}}^{m}{\lambda }_{i}{a}_{i} \), where \( \mathop{\sum }\limits_{{i... | ## Proof. The proof is straightforward and is omitted. It is also given in Gallier [71]. | No |
Proposition 24.6. Given an affine space \( \langle E,\overrightarrow{E}, + \rangle \), let \( {\left( {a}_{i}\right) }_{i \in I} \) be a family of points in \( E \) . The family \( {\left( {a}_{i}\right) }_{i \in I} \) is affinely dependent iff there is a family \( {\left( {\lambda }_{i}\right) }_{i \in I} \) such that... | Proof. By Proposition 24.5, the family \( {\left( {a}_{i}\right) }_{i \in I} \) is affinely dependent iff the family of vectors \( {\left( \overrightarrow{{a}_{i}{a}_{j}}\right) }_{j \in \left( {I-\{ i\} }\right) } \) is linearly dependent for some \( i \in I \) . For any \( i \in I \), the family \( {\left( \overright... | Yes |
Proposition 24.7. Given any point \( a \in E \), any point \( b \in {E}^{\prime } \), and any linear map \( h : \overrightarrow{E} \rightarrow \) \( \overrightarrow{{E}^{\prime }} \), the map \( f : E \rightarrow {E}^{\prime } \) defined such that\n\n\[ f\left( {a + v}\right) = b + h\left( v\right) \]\n\nis an affine m... | Proof. Indeed, for any family \( {\left( {\lambda }_{i}\right) }_{i \in I} \) of scalars with \( \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i} = 1 \) and any family \( {\left( {v}_{i}\right) }_{i \in I} \), since\n\n\[ \mathop{\sum }\limits_{{i \in I}}{\lambda }_{i}\left( {a + {v}_{i}}\right) = a + \mathop{\sum }\lim... | Yes |
Proposition 24.9. Given any affine space \( E \), for any affine bijection \( f \in \mathbf{{GA}}\left( E\right) \), if \( \overrightarrow{f} = \) \( \lambda {\operatorname{id}}_{\overrightarrow{E}} \), for some \( \lambda \in {\mathbb{R}}^{ * } \) with \( \lambda \neq 1 \), then there is a unique point \( c \in E \) s... | Proof. The proof is straightforward, and is omitted. It is also given in Gallier [71]. | No |
Proposition 24.11. Given any affine space \( E \), given any two distinct points \( a, b \in E \), and for any affine dilatation \( f \) different from the identity, if \( {a}^{\prime } = f\left( a\right), D = \langle a, b\rangle \) is the line passing through a and \( b \), and \( {D}^{\prime } \) is the line parallel... | The first case is the parallelogram law, and the second case follows easily from Thales' theorem. For an illustration, see Figure 24.22. | No |
Given any affine plane \( E \), any two distinct lines \( D \) and \( {D}^{\prime } \), then for any distinct points \( a, b, c \) on \( D \) and \( {a}^{\prime },{b}^{\prime },{c}^{\prime } \) on \( {D}^{\prime } \), if \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \) are distinct from the intersection of \( D ... | Proof. Pappus’s theorem is illustrated in Figure 24.23. If \( D \) and \( {D}^{\prime } \) are not parallel, let \( d \) be their intersection. Let \( f \) be the dilatation of center \( d \) such that \( f\left( a\right) = b \), and let \( g \) be the dilatation of center \( d \) such that \( g\left( b\right) = c \) .... | Yes |
Proposition 24.13. Given any affine space \( E \), and given any two triangles \( \left( {a, b, c}\right) \) and \( \left( {{a}^{\prime },{b}^{\prime },{c}^{\prime }}\right) \), where \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \) are all distinct, if \( \langle a, b\rangle \) and \( \left\langle {{a}^{\prime ... | Proof. We prove half of the proposition, the direction in which it is assumed that \( \langle a, c\rangle \) and \( \left\langle {{a}^{\prime },{c}^{\prime }}\right\rangle \) are parallel, leaving the converse as an exercise. Since the lines \( \langle a, b\rangle \) and \( \left\langle {{a}^{\prime },{b}^{\prime }}\ri... | No |
Proposition 24.14. Let \( E \) be an affine space. The following properties hold:\n\n(a) Given any nonconstant affine form \( f : E \rightarrow \mathbb{R} \), its kernel \( H = \operatorname{Ker}f \) is a hyperplane.\n\n(b) For any hyperplane \( H \) in \( E \), there is a nonconstant affine form \( f : E \rightarrow \... | Proof. The proof is straightforward, and is omitted. It is also given in Gallier [71]. | No |
Proposition 24.15. Given a vector space \( E \) and any two subspaces \( M \) and \( N \), with the definitions above,\n\n\[ 0 \rightarrow M \cap N\overset{f + g}{ \rightarrow }M \oplus N\overset{i - j}{ \rightarrow }M + N \rightarrow 0 \]\n\nis a short exact sequence, which means that \( f + g \) is injective, \( i - ... | Proof. It is obvious that \( i - j \) is surjective and that \( f + g \) is injective. Assume that ( \( i - \) \( j)\left( {u + v}\right) = 0 \), where \( u \in M \), and \( v \in N \) . Then, \( i\left( u\right) = j\left( v\right) \), and thus, by definition of \( i \) and \( j \), there is some \( w \in M \cap N \), ... | Yes |
Proposition 24.16. Given any affine space \( E \) , for any two nonempty affine subspaces \( M \) and \( N \), the following facts hold:\n\n(1) \( M \cap N \neq \varnothing \) iff \( \overrightarrow{ab} \in \overrightarrow{M} + \overrightarrow{N} \) for some \( a \in M \) and some \( b \in N \) . | Proof. (1) Pick any \( a \in M \) and any \( b \in N \), which is possible, since \( M \) and \( N \) are nonempty. Since \( \overrightarrow{M} = \{ \overrightarrow{ax} \mid x \in M\} \) and \( \overrightarrow{N} = \{ \overrightarrow{by} \mid y \in N\} \), if \( M \cap N \neq \varnothing \), for any \( c \in M \cap N \... | Yes |
Proposition 25.2. Given any affine space \( \left( {E,\overrightarrow{E}}\right) \), for any family \( {\left( {a}_{i}\right) }_{i \in I} \) of points in \( E \) , any family \( {\left( {\lambda }_{i}\right) }_{i \in I} \) of scalars in \( \mathbb{R} \), and any family \( {\left( {v}_{j}\right) }_{j \in J} \) of vector... | Proof. By induction on the size of \( I \) and the size of \( J \) . | No |
Proposition 25.3. Given any affine space \( \left( {E,\overrightarrow{E}}\right) \), for any affine frame \( \left( {{a}_{0},\left( {\overrightarrow{{a}_{0}{a}_{1}},\ldots }\right. }\right. \) , \( \left. \overrightarrow{\left. {a}_{0}{a}_{m}\right) }\right) \) for \( E \), the family \( \left( {\overrightarrow{{a}_{0}... | Proof. We sketch parts of the proof, leaving the details as an exercise. Figure 25.2 shows the basis \( \left( {\overrightarrow{{a}_{0}{a}_{1}},\overrightarrow{{a}_{0}{a}_{2}},{a}_{0}}\right) \) corresponding to the affine frame \( \left( {{a}_{0},{a}_{1},{a}_{2}}\right) \) in \( E \) . | No |
Given any affine space \( \left( {E,\overrightarrow{E}}\right) \), for any choice \( {\Omega }_{1} \) of an origin in \( E \), the map \( \widehat{\Omega } : \widehat{E} \rightarrow \mathcal{F} \) is a linear isomorphism between \( \widehat{E} \) and the vector space \( \mathcal{F} \) of Definition 25.1. The inverse of... | \[ {\widehat{\Omega }}^{-1}\left( {u + ╏}\right) = \left\{ \begin{array}{ll} \left\langle {{\Omega }_{1} + {\lambda }^{-1}u,\lambda }\right\rangle ) & \text{ if }\lambda \neq 0 \\ u & \text{ if }\lambda = 0 \end{array}\right. \] Proof. It is a straightforward verification. We check that \( \widehat{\Omega } \) is inver... | No |
Proposition 25.5. Given any affine space \( \left( {E,\overrightarrow{E}}\right) \) and any vector space \( \overrightarrow{F} \) , for any affine map \( f : E \rightarrow \overrightarrow{F} \), there is a unique linear map \( \widehat{f} : \widehat{E} \rightarrow \overrightarrow{F} \) extending \( f \) such that\n\n\[... | Proof. Assuming that \( \widehat{f} \) exists, recall that from Proposition 25.1, for every \( a \in E \), every element of \( \widehat{E} \) can be written uniquely as \( u\widehat{ + }{\lambda a} \) . By linearity of \( \widehat{f} \) and since \( \widehat{f} \) extends \( f \), we have\n\n\[ \widehat{f}\left( {u\wid... | Yes |
Proposition 25.6. Given two affine spaces \( E \) and \( F \) and an affine map \( f : E \rightarrow F \), there is a unique linear map \( \widehat{f} : \widehat{E} \rightarrow \widehat{F} \) extending \( f \), as in the diagram below, such that \[ \widehat{f}\left( {u\widehat{ + }{\lambda a}}\right) = \overrightarrow{... | Proof. Consider the vector space \( \widehat{F} \) and the affine map \( j \circ f : E \rightarrow \widehat{F} \) . By Proposition 25.5, there is a unique linear map \( \widehat{f} : \widehat{E} \rightarrow \widehat{F} \) extending \( j \circ f \), and thus extending \( f \). Note that \( \widehat{f} : \widehat{E} \rig... | Yes |
Proposition 26.2. If \( {\left( {a}_{i}\right) }_{1 \leq i \leq n + 2} \) is a projective frame of \( \mathbf{P}\left( E\right) \), for any two bases \( \left( {{u}_{1},\ldots }\right. \) , \( \left. {u}_{n + 1}\right) ,\left( {{v}_{1},\ldots ,{v}_{n + 1}}\right) \) of \( E \) such that \( {a}_{i} = p\left( {u}_{i}\rig... | Proof. Since \( p\left( {u}_{i}\right) = p\left( {v}_{i}\right) \) for \( 1 \leq i \leq n + 1 \), there exist some nonzero scalars \( {\lambda }_{i} \in K \) such that \( {v}_{i} = {\lambda }_{i}{u}_{i} \) for all \( i,1 \leq i \leq n + 1 \) . Since we must have\n\n\[ p\left( {{u}_{1} + \cdots + {u}_{n + 1}}\right) = p... | Yes |
Proposition 26.3. A family \( {\left( {a}_{i}\right) }_{1 \leq i \leq n + 2} \) of \( n + 2 \) points is a projective frame of \( \mathbf{P}\left( E\right) \) iff for every \( i,1 \leq i \leq n + 2 \), the subfamily \( {\left( {a}_{j}\right) }_{j \neq i} \) is projectively independent. | Proof. We leave as an (easy) exercise the fact that if \( {\left( {a}_{i}\right) }_{1 \leq i \leq n + 2} \) is a projective frame, then each subfamily \( {\left( {a}_{j}\right) }_{j \neq i} \) is projectively independent. Conversely, pick some \( {u}_{i} \in E - \{ 0\} \) such that \( {a}_{i} = p\left( {u}_{i}\right) ,... | No |
Proposition 26.5. Given two nontrivial vector spaces \( E \) and \( F \) of the same dimension \( n + 1 \), for any two projective frames \( {\left( {a}_{i}\right) }_{1 \leq i \leq n + 2} \) for \( \mathbf{P}\left( E\right) \) and \( {\left( {b}_{i}\right) }_{1 \leq i \leq n + 2} \) for \( \mathbf{P}\left( F\right) \),... | Proof. Let \( \left( {{u}_{1},\ldots ,{u}_{n + 1}}\right) \) be a basis of \( E \) associated with the projective frame \( {\left( {a}_{i}\right) }_{1 \leq i \leq n + 2} \) , and let \( \left( {{v}_{1},\ldots ,{v}_{n + 1}}\right) \) be a basis of \( F \) associated with the projective frame \( {\left( {b}_{i}\right) }_... | Yes |
Proposition 26.6. Given any two distinct lines \( D \) and \( {D}^{\prime } \) in the real projective plane \( {\mathbb{{RP}}}^{2} \) , a projectivity \( f : D \rightarrow {D}^{\prime } \) is a perspectivity iff \( f\left( O\right) = O \), where \( O \) is the intersection of \( D \) and \( {D}^{\prime } \) . | Proof. If \( f : D \rightarrow {D}^{\prime } \) is a perspectivity, then by the very definition of \( f \), we have \( f\left( O\right) = O \) . Conversely, let \( f : D \rightarrow {D}^{\prime } \) be a projectivity such that \( f\left( O\right) = O \) . Let \( a, b \) be any two distinct points on \( D \) also distin... | Yes |
Proposition 26.7. (Desargues) Given two triangles \( \left( {a, b, c}\right) \) and \( \left( {{a}^{\prime },{b}^{\prime },{c}^{\prime }}\right) \) in \( {\mathbb{{RP}}}^{2} \), where the points \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \) are pairwise distinct and the lines \( A = \langle b, c\rangle, B = \... | Proof. In view of the assumptions on \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \), and \( d \), the point \( r \) is on neither \( \left\langle {a,{a}^{\prime }}\right\rangle \) nor \( \left\langle {b,{b}^{\prime }}\right\rangle \), the point \( p \) is on neither \( \left\langle {b,{b}^{\prime }}\right\rang... | Yes |
Proposition 26.8. With respect to the basis \( \mathcal{P} = \left( {{p}_{1},{p}_{2},{p}_{3}}\right) \), the matrix \( {A}_{\mathcal{P}} \) of the unique homography \( h \) of \( {\mathbb{{RP}}}^{2} \) mapping the projective frame \( \left( {\left\lbrack {p}_{1}\right\rbrack ,\left\lbrack {p}_{2}\right\rbrack ,\left\lb... | Proof. Let \( {u}_{1} = {\alpha }_{1}{p}_{1},{u}_{2} = {\alpha }_{2}{p}_{2},{u}_{3} = {\alpha }_{3}{p}_{3} \), and let \( {v}_{1} = {\lambda }_{1}{q}_{1},{v}_{2} = {\lambda }_{2}{q}_{2},{v}_{3} = {\lambda }_{3}{q}_{3} \), so that \[ {p}_{4} = {u}_{1} + {u}_{2} + {u}_{3} \] and \[ {q}_{4} = {v}_{1} + {v}_{2} + {v}_{3} \... | Yes |
Proposition 26.11. With respect to the canonical basis \( \mathcal{E} = \left( {{e}_{1},\ldots ,{e}_{n + 1}}\right) \), the matrix \( {A}_{\mathcal{E}} \) of the unique homography \( h \) of \( \mathbb{P}\left( E\right) \) where \( E \) is a \( K \) -vector space of dimension \( n + 1 \), mapping the projective frame \... | \[ {A}_{\mathcal{E}} = \left( \begin{matrix} {q}_{1}^{1} & \ldots & {q}_{n}^{1} & {q}_{n + 1}^{1} \\ \vdots & \ddots & \vdots & \vdots \\ {q}_{1}^{n} & \ldots & {q}_{n}^{n} & {q}_{n + 1}^{n} \\ {q}_{1}^{n + 1} & \ldots & {q}_{n}^{n + 1} & {q}_{n + 1}^{n + 1} \end{matrix}\right) \left( \begin{matrix} \frac{{\lambda }_{1... | Yes |
Proposition 26.12. With respect to the canonical basis \( \mathcal{E} = \left( {{e}_{1},{e}_{2},{e}_{3}}\right) \), the matrix \( {A}_{\mathcal{E}} \) of the unique homography \( h \) of \( {\mathbb{{RP}}}^{2} \) mapping \( \left( {{p}_{1},{p}_{2},{p}_{4},{p}_{4}}\right) \), points of the affine plane \( z = 1 \), to \... | \[ {A}_{\mathcal{E}} = \left( \begin{array}{lll} {q}_{1}^{x} & {q}_{2}^{x} & {q}_{3}^{x} \\ {q}_{1}^{y} & {q}_{2}^{y} & {q}_{3}^{y} \\ {q}_{1}^{z} & {q}_{2}^{z} & {q}_{3}^{z} \end{array}\right) \left( \begin{matrix} \frac{{\lambda }_{1}}{{\alpha }_{1}} & 0 & 0 \\ 0 & \frac{{\lambda }_{2}}{{\alpha }_{2}} & 0 \\ 0 & 0 & ... | Yes |
Proposition 26.13. With respect to the canonical basis \( \mathcal{E} = \left( {{e}_{1},{e}_{2},{e}_{3}}\right) \), the matrix \( {A}_{\mathcal{E}} \) of the unique homography \( h \) of \( {\mathbb{{RP}}}^{2} \) mapping \( \left( {{p}_{1},{p}_{2},{p}_{4},{p}_{4}}\right) \) to \( \left( {{q}_{1},{q}_{2},{q}_{3},{q}_{4}... | \[ {A}_{\mathcal{E}} = \left( \begin{matrix} {q}_{1}^{x} & {q}_{2}^{x} & {q}_{3}^{x} \\ {q}_{1}^{y} & {q}_{2}^{y} & {q}_{3}^{y} \\ 1 & 1 & 1 \end{matrix}\right) \left( \begin{matrix} \frac{{\lambda }_{1}}{{\alpha }_{1}} & 0 & 0 \\ 0 & \frac{{\lambda }_{2}}{{\alpha }_{2}} & 0 \\ 0 & 0 & \frac{{\lambda }_{3}}{{\alpha }_{... | Yes |
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