Matrix properties and the smallest eigenvalue

Asked Aug 19 '13 at 20:06

Active Aug 19 '13 at 20:58

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Let $\lambda_{\min}$ be the smallest eigenvalue of the positive definite matrix $\mathbf{S}$, and $\|\mathbf{a}\|=r$. Then $$ \mathbf{a}^T \mathbf{S}\mathbf{a} > \frac{1}{2}\lambda_{\min} r^2 $$

What properties of matrices were used to obtain the result? Thanks!

edited Aug 19 '13 at 20:58

dtldarek

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asked Aug 19 '13 at 20:06

Mandy

Are you aware of Minimum and Maximum eigenvalue inequality from a positive definite matrix? – M Turgeon Aug 19 '13 at 20:27
You can find orthogonal matrix $U$ such that $U^tSU$ is diagonal with the eigenvalues of $S$ is the diagonal. – OR. Aug 19 '13 at 20:29
Thanks @MTurgeon, that gives $\mathbf{a}^T\mathbf{S}\mathbf{a} \geq \lambda_{min}r^2$, where does the half come in? – Mandy Aug 19 '13 at 20:55
1

@Mandy $\alpha\geq \frac{1}{2}\alpha$ for all non-negative $\alpha$. – M Turgeon Aug 19 '13 at 21:21

Matrix properties and the smallest eigenvalue

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