samples/texts/227520/page_13.md

which shows that

$$I_b+U(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U(4.2)I\_b\; +\; U\; (\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U\; \backslash quad\; (4.2)$$

is a singular matrix. The singularity of (4.2) implies

i. e., a QR factorization with column pivoting (Golub and Van Loan [17]) with $1 \le m \le b$, an orthonormal matrix $Q_i \in \mathbb{R}^{b \times b}$, a permutation matrix $P_i \in \mathbb{R}^{b \times b}$ (i. e., $Q_i Q_i = P_i P_i = I_b$), a nonsingular upper triangular matrix $R_{i1} \in \mathbb{R}^{(b \ m_1) \times (b \ m_1)}$, and a matrix $R_{i2} \in \mathbb{R}^{(b \ m_2) \times (b \ m_2)}$. The matrix

$$X_i:=P_i\left(\begin{array}{c}-R_{i1}^{\mathrm{\top }}{R}_{i2}\\ I_m\end{array}\right)\in {\mathbb{R}}^{b\times m}(4.4)X\_i\; :=\; P\_i\; \backslash begin\{pmatrix\}\; -R\_\{i1\}^\{\backslash top\}\; R\_\{i2\}\; \backslash \backslash \; I\_m\; \backslash end\{pmatrix\}\; \backslash in\; \backslash mathbb\{R\}^\{b\; \backslash times\; m\}\; \backslash quad\; (4.4)$$

has rank m and (because of (4.3)) its columns span the null space of (4.2), i. e.,

$$(I_b+U(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U)X_i=0.(4.5)(I\_b\; +\; U\; (\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U)\; X\_i\; =\; 0.\; \backslash quad\; (4.5)$$

Computation of the Eigenvectors. Compute the QR factorization of the $n \times m$ matrix

$$(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U{X}_i=Z_i{S}_i(4.6)(\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U\; X\_i\; =\; Z\_i\; S\_i\; \backslash quad\; (4.6)$$

with $Z_i \in \mathbb{R}^{n \times m}$ and nonsingular upper triangular $S_i \in \mathbb{R}^{m \times m}$. The column vectors of $Z_i$ form a set of $l$ orthonormal eigenvectors of $S$. From

it follows that $(\Sigma + UU) Z_i = \sigma_i^* Z_i$, i. e., the columns of $Z_i$ are eigenvectors of $S$ corresponding to $\sigma_i^*$. They are orthonormal by construction.

It remains to be shown that all $l$ eigenvectors corresponding to the eigenvalue $\sigma_i^*$ have been found, that is, $m = l$. First, since the geometric multiplicity of an eigenvalue is less or equal its algebraic multiplicity (Horn and Johnson [19]), $m \le l$. Equality can be shown indirectly. Assume that $m < l$. Then there must be another eigenvector $z_i$ of $S$ with $|z_i|_2 = 1$, $z_i^T Z_i z_i = 0$, and, according to (4.1),

$$(\mathrm{\Sigma }+UU-{\sigma }_i^{\ast }{I}_n)z_i=0(4.7)(\backslash Sigma\; +\; UU\; -\; \backslash sigma\_i^*\; I\_n)\; z\_i\; =\; 0\; \backslash quad\; (4.7)$$

$$(I_b+U(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U)U{z}_i=0.(I\_b\; +\; U\; (\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U)\; U\; z\_i\; =\; 0.$$

This implies that $U z_i$ lies in the null space of

$$I_b+U(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U,I\_b\; +\; U\; (\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U,$$

and since $X_i$ is a basis of this null space (according to (4.5)), there exists a vector $y \in \mathbb{R}^m$ such that $U z_i = X_i y$. Thus

$$(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}UU{z}_i=(\mathrm{\Sigma }-{\sigma }_i^{\ast }{I}_n{)}^{-1}U{X}_{i}y=(4.6)=Z_i{S}_{i}y\mathrm{.}(\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U\; U\; z\_i\; =\; (\backslash Sigma\; -\; \backslash sigma\_i^*\; I\_n)^\{-1\}\; U\; X\_i\; y\; =\; (4.6)\; =\; Z\_i\; S\_i\; y.$$