Editing Singular value (section)

== Inequalities about singular values ==
See also.<ref>[[Roger Horn|R. A. Horn]] and [[Charles Royal Johnson|C. R. Johnson]]. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. Chap. 3</ref>

===Singular values of sub-matrices===

For <math>A \in \mathbb{C}^{m \times n}.</math>
# Let <math>B</math> denote <math>A</math> with one of its rows ''or'' columns deleted. Then <math display="block">\sigma_{i+1}(A) \leq \sigma_i (B) \leq \sigma_i(A)</math>
# Let <math>B</math> denote <math>A</math> with one of its rows ''and'' columns deleted. Then <math display="block">\sigma_{i+2}(A) \leq \sigma_i (B) \leq \sigma_i(A)</math>
# Let <math>B</math> denote an <math>(m-k)\times(n-\ell)</math> submatrix of <math>A</math>. Then <math display="block">\sigma_{i+k+\ell}(A) \leq \sigma_i (B) \leq \sigma_i(A)</math>

===Singular values of ''A'' + ''B''===

For <math>A, B \in \mathbb{C}^{m \times n}</math>
# <math display="block">\sum_{i=1}^{k} \sigma_i(A + B) \leq \sum_{i=1}^{k} (\sigma_i(A) + \sigma_i(B)), \quad k=\min \{m,n\}</math>
# <math display="block">\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\}</math>

===Singular values of ''AB''===

For <math>A, B \in \mathbb{C}^{n \times n}</math>
# <math display="block">\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}</math>
# <math display="block">\sigma_n(A) \sigma_i(B) \leq \sigma_i (AB) \leq \sigma_1(A) \sigma_i(B) \quad i = 1, 2, \ldots, n. </math>

For <math>A, B \in \mathbb{C}^{m \times n}</math><ref>X. Zhan. Matrix Inequalities. Springer-Verlag, Berlin, Heidelberg, 2002. p.28</ref>
<math display="block">2 \sigma_i(A B^*) \leq \sigma_i \left(A^* A + B^* B\right), \quad i = 1, 2, \ldots, n. </math>

===Singular values and eigenvalues===

For <math>A \in \mathbb{C}^{n \times n}</math>.
# See<ref>R. Bhatia. Matrix Analysis. Springer-Verlag, New York, 1997. Prop. III.5.1</ref> <math display="block">\lambda_i \left(A + A^*\right) \leq 2 \sigma_i(A), \quad i = 1, 2, \ldots, n.</math>
# Assume <math>\left|\lambda_1(A)\right| \geq \cdots \geq \left|\lambda_n(A)\right|</math>. Then for <math>k = 1, 2, \ldots, n</math>:
## [[Weyl's inequality#Weyl's inequality in matrix theory|Weyl's theorem]] <math display="block"> \prod_{i=1}^k \left|\lambda_i(A)\right| \leq \prod_{i=1}^{k} \sigma_i(A).</math>
## For <math>p>0</math>. <math display="block"> \sum_{i=1}^k \left|\lambda_i^p(A)\right| \leq \sum_{i=1}^{k} \sigma_i^p(A).</math>