Editing Matrix norm (section)

==== Spectral norm (''p'' = 2) ====
{{anchor|Spectral norm}}
When <math>p = 2</math> (the [[Euclidean norm]] or <math>\ell_2</math>-norm for vectors), the induced matrix norm is the ''spectral norm''. The two values do ''not'' coincide in infinite dimensions &mdash; see [[Spectral radius]] for further discussion. The spectral radius should not be confused with the spectral norm. The spectral norm of a matrix <math>A</math> is the largest [[singular value]] of <math>A</math>, i.e., the square root of the largest [[eigenvalue]] of the matrix <math>A^*A,</math> where <math>A^*</math> denotes the [[conjugate transpose]] of <math>A</math>:<ref>Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, §5.2, p.281, Society for Industrial & Applied Mathematics, June 2000.</ref><math display="block"> \|A\|_2 = \sqrt{\lambda_{\max}\left(A^* A\right)} = \sigma_{\max}(A).</math>where <math>\sigma_{\max}(A)</math> represents the largest singular value of matrix <math>A.</math>

There are further properties:
* <math display="inline">\|A \|_2 = \sup\{x^* A y : x \in K^m, y \in K^n \text{ with }\|x\|_2 = \|y\|_2 = 1\}.</math> Proved by the [[Cauchy–Schwarz inequality]].
* <math display="inline"> \| A^* A\|_2 = \| A A^* \|_2 = \|A\|_2^2</math>. Proven by [[singular value decomposition]] (SVD) on <math>A</math>.
* <math display="inline"> \|A\| _2 = \sigma_{\mathrm{max}}(A) \leq \|A\|_{\rm F} = \sqrt{\sum_i \sigma_{i}(A)^2}</math>, where <math>\|A\|_\textrm{F}</math> is the [[#Frobenius norm|Frobenius norm]]. Equality holds if and only if the matrix <math>A</math> is a rank-one matrix or a zero matrix.
* Conversely, <math>\|A\|_\textrm{F} \leq \min(m,n)^{1/2}\|A\|_2</math>.

* <math> \|A\|_2 = \sqrt{\rho(A^{*}A)}\leq\sqrt{\|A^{*}A\|_\infty}\leq\sqrt{\|A\|_1\|A\|_\infty} </math>.