Editing Singular value (section)

== Basic properties ==

For <math>A \in \mathbb{C}^{m \times n}</math>, and <math>i = 1,2, \ldots, \min \{m,n\}</math>.

[[Min-max theorem#Min-max principle for singular values|Min-max theorem for singular values]]. Here <math>U: \dim(U) = i</math> is a subspace of <math>\mathbb{C}^n</math> of dimension <math>i</math>.

:<math>\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}</math>

Matrix transpose and conjugate do not alter singular values.

:<math>\sigma_i(A) = \sigma_i\left(A^\textsf{T}\right) = \sigma_i\left(A^*\right).</math>

For any unitary <math>U \in \mathbb{C}^{m \times m}, V \in \mathbb{C}^{n \times n}.</math>

:<math>\sigma_i(A) = \sigma_i(UAV).</math>

Relation to eigenvalues:

:<math>\sigma_i^2(A) = \lambda_i\left(AA^*\right) = \lambda_i\left(A^*A\right).</math>

Relation to [[Trace (linear algebra)|trace]]:

:<math>\sum_{i=1}^n \sigma_i^2=\text{tr}\ A^\ast A</math>.

If <math>A^*  A</math> is full rank, the product of singular values is <math>\det \sqrt{A^* A}</math>.

If <math>A A^*</math> is full rank, the product of singular values is <math>\det\sqrt{ A A^*}</math>.

If <math>A</math> is square and full rank, the product of singular values is <math>|\det A|</math>.

If <math>A</math> is [[Normal matrix|normal]], then <math>\sigma(A) = |\lambda(A)|</math>, that is, its singular values are the absolute values of its eigenvalues.

For a generic rectangular matrix <math>A</math>, let <math display="inline">\tilde{A} = \begin{bmatrix} 0 & A \\ A^* & 0 \end{bmatrix}</math> be its augmented matrix. It has eigenvalues <math display="inline">\pm \sigma(A)</math> (where <math display="inline">\sigma(A)</math> are the singular values of <math display="inline">A</math>) and the remaining eigenvalues are zero. Let <math display="inline">A = U\Sigma V^*</math> be the singular value decomposition, then the eigenvectors of <math display="inline">\tilde{A}</math> are <math display="inline">\begin{bmatrix} \mathbf{u}_i \\ \pm\mathbf{v}_i \end{bmatrix}</math> for <math>\pm  \sigma_i</math><ref>{{Cite book |last=Tao |first=Terence |title=Topics in random matrix theory |date=2012 |publisher=American Mathematical Society |isbn=978-0-8218-7430-1 |series=Graduate studies in mathematics |location=Providence, R.I}}</ref>{{Pg|page=52}}