Singular value

For A \in \mathbb{C}^{m \times n}, and i = 1,2, \ldots, \min \{m,n\}. Min-max theorem for singular values. Here U: \dim(U) = i is a subspace of \mathbb{C}^n of dimension i. :\begin{align} \sigma_i(A) &= \min_{\dim(U)=n-i+1} \max_{\underset{\| x \|_2 = 1}{x \in U}} \left\| Ax \right\|_2. \\ \sigma_i(A) &= \max_{\dim(U)=i} \min_{\underset{\| x \|_2 = 1}{x \in U}} \left\| Ax \right\|_2. \end{align} Matrix transpose and conjugate do not alter singular values. :\sigma_i(A) = \sigma_i\left(A^\textsf{T}\right) = \sigma_i\left(A^*\right). For any unitary U \in \mathbb{C}^{m \times m}, V \in \mathbb{C}^{n \times n}. :\sigma_i(A) = \sigma_i(UAV). Relation to eigenvalues: :\sigma_i^2(A) = \lambda_i\left(AA^*\right) = \lambda_i\left(A^*A\right). Relation to trace: :\sum_{i=1}^n \sigma_i^2=\text{tr}\ A^\ast A. If A^* A has full rank, the product of singular values is \det \sqrt{A^* A}. If A A^* has full rank, the product of singular values is \det\sqrt{ A A^*}. If A is square and has full rank, the product of singular values is |\det A|. If A is normal, then \sigma(A) = |\lambda(A)|, that is, its singular values are the absolute values of its eigenvalues. For a generic rectangular matrix A, let \tilde{A} = \begin{bmatrix} 0 & A \\ A^* & 0 \end{bmatrix} be its augmented matrix. It has eigenvalues \pm \sigma(A) (where \sigma(A) are the singular values of A) and the remaining eigenvalues are zero. Let A = U\Sigma V^* be the singular value decomposition, then the eigenvectors of \tilde{A} are \begin{bmatrix} \mathbf{u}_i \\ \pm\mathbf{v}_i \end{bmatrix} for \pm \sigma_i == The smallest singular value ==

The smallest singular value of a matrix A is σn(A). It has the following properties for a non-singular matrix A: • The 2-norm of the inverse matrix A−1 equals the inverse σn−1(A). • The absolute values of all elements in the inverse matrix A−1 are at most the inverse σn−1(A). Intuitively, if σn(A) is small, then the rows of A are "almost" linearly dependent. If it is σn(A) = 0, then the rows of A are linearly dependent and A is not invertible. == Inequalities about singular values ==

See also. Singular values of sub-matrices For A \in \mathbb{C}^{m \times n}. • Let B denote A with one of its rows or columns deleted. Then \sigma_{i+1}(A) \leq \sigma_i (B) \leq \sigma_i(A) • Let B denote A with two of its rows and columns deleted. Then \sigma_{i+2}(A) \leq \sigma_i (B) \leq \sigma_i(A) • Let B denote an (m-k)\times(n-\ell) submatrix of A. Then \sigma_{i+k+\ell}(A) \leq \sigma_i (B) \leq \sigma_i(A) Singular values of A + B For A, B \in \mathbb{C}^{m \times n} • \sum_{i=1}^{k} \sigma_i(A + B) \leq \sum_{i=1}^{k} (\sigma_i(A) + \sigma_i(B)), \quad k=\min \{m,n\} • \sigma_{i+j-1}(A + B) \leq \sigma_i(A) + \sigma_j(B). \quad i,j\in\mathbb{N},\ i + j - 1 \leq \min \{m,n\} Singular values of AB For A, B \in \mathbb{C}^{n \times n} • \begin{align} \prod_{i=n}^{i=n-k+1} \sigma_i(A) \sigma_i(B) &\leq \prod_{i=n}^{i=n-k+1} \sigma_i(AB) \\ \prod_{i=1}^k \sigma_i(AB) &\leq \prod_{i=1}^k \sigma_i(A) \sigma_i(B), \\ \sum_{i=1}^k \sigma_i^p(AB) &\leq \sum_{i=1}^k \sigma_i^p(A) \sigma_i^p(B), \end{align} • \sigma_n(A) \sigma_i(B) \leq \sigma_i (AB) \leq \sigma_1(A) \sigma_i(B) \quad i = 1, 2, \ldots, n. For A, B \in \mathbb{C}^{m \times n} 2 \sigma_i(A B^*) \leq \sigma_i \left(A^* A + B^* B\right), \quad i = 1, 2, \ldots, n. Singular values and eigenvalues For A \in \mathbb{C}^{n \times n}. • See \lambda_i \left(A + A^*\right) \leq 2 \sigma_i(A), \quad i = 1, 2, \ldots, n. • Assume \left|\lambda_1(A)\right| \geq \cdots \geq \left|\lambda_n(A)\right|. Then for k = 1, 2, \ldots, n: • Weyl's theorem \prod_{i=1}^k \left|\lambda_i(A)\right| \leq \prod_{i=1}^{k} \sigma_i(A). • For p>0. \sum_{i=1}^k \left|\lambda_i^p(A)\right| \leq \sum_{i=1}^{k} \sigma_i^p(A). == History ==

This concept was introduced by Erhard Schmidt in 1907. Schmidt called singular values "eigenvalues" at that time. The name "singular value" was first quoted by Smithies in 1937. In 1957, Allahverdiev proved the following characterization of the nth singular number: : \sigma_n(T) = \inf\big\{\, \|T-L\| : L\text{ is an operator of finite rank } This formulation made it possible to extend the notion of singular values to operators in Banach space. Note that there is a more general concept of s-numbers, which also includes Gelfand and Kolmogorov width. == See also ==