Editing Spectral radius (section)

==Symmetric matrices==

For real-valued matrices <math>A</math> the inequality <math>\rho(A) \leq {\|A\|}_{2}</math> holds in particular, where <math>{\|\cdot\|}_{2}</math> denotes the [[Matrix_norm#Spectral_norm|spectral norm]]. In the case
where <math>A</math> is [[:en:Symmetric_matrix|symmetric]], this inequality is tight:

'''Theorem.''' Let <math>A \in \mathbb{R}^{n \times n}</math> be symmetric, i.e., <math>A = A^T.</math> Then it holds that <math>\rho(A) = {\|A\|}_{2}.</math>

'''Proof'''

Let <math>(v_i, \lambda_i)_{i=1}^{n}</math> be the eigenpairs of ''A''. Due to the symmetry of ''A'',
all <math>v_i</math> and <math>\lambda_i</math> are real-valued and the eigenvectors <math>v_i</math> are [[Orthonormal_basis|orthonormal]].
By the definition of the spectral norm, there exists an <math>x \in \mathbb{R}^{n}</math> with
<math>{\|x\|}_{2} = 1</math> such that <math>{\|A\|}_{2} = {\| A x \|}_{2}.</math> Since the eigenvectors <math>v_i</math> form a basis
of <math>\mathbb{R}^{n},</math> there exists
factors <math>\delta_{1}, \ldots, \delta_{n} \in \mathbb{R}^{n}</math> such that <math>\textstyle x = \sum_{i = 1}^{n} \delta_{i} v_{i}</math> which implies that
:<math>A x = \sum_{i = 1}^{n}\delta_{i} A v_{i} = \sum_{i = 1}^{n} \delta_{i} \lambda_{i} v_{i}.</math>

From the orthonormality of the eigenvectors <math>v_i</math> it follows that
:<math>{\| A x\|}_{2} = \| \sum_{i = 1}^{n} \delta_{i} \lambda_{i} v_{i}\|_{2} = \sum_{i = 1}^{n} {|\delta_{i}|} \cdot {|\lambda_{i}|} \cdot {\| v_{i}\|}_{2} = \sum_{i = 1}^{n} {|\delta_{i}|} \cdot {|\lambda_{i}|} </math>
and 
:<math>{\|x\|}_{2} = \| \sum_{i = 1}^{n} \delta_{i} v_{i} \|_{2} = \sum_{i = 1}^{n} {|\delta_{i}|} \cdot {\| v_{i} \|}_{2} = \sum_{i = 1}^{n} {|\delta_{i}|}.</math>

Since <math>x</math> is chosen such that it maximizes <math>{\|Ax\|}_{2}</math> while satisfying <math>{\|x\|}_{2} = 1,</math> the values of <math>\delta_{i}</math> must be such that they maximize <math>\textstyle \sum_{i = 1}^{n} {|\delta_{i}|} \cdot {|\lambda_{i}|}</math> while satisfying <math>\textstyle \sum_{i = 1}^{n} {|\delta_{i}|} = 1.</math> This is achieved by setting <math>\delta_{k} = 1</math> for <math>k = \mathrm{arg\,max}_{i=1}^{n} {|\lambda_i|}</math> and <math>\delta_{i} = 0</math> otherwise, yielding a value of <math>{\|Ax\|}_{2} = {|\lambda_k|} = \rho(A).</math>