Matrix properties and the smallest eigenvalue [duplicate]

Matrix properties and the smallest eigenvalue [duplicate]

div class=question-status question-originals-of-duplicate pThis question
already has an answer here:/p ul li a
href=/questions/245866/minimum-and-maximum-eigenvalue-inequality-from-a-positive-definite-matrix
dir=ltrMinimum and Maximum eigenvalue inequality from a positive definite
matrix./a span class=question-originals-answer-count 2 answers /span /li
/ul /div pLet $\lambda_{\min}$ be the smallest eigenvalue of the positive
definite matrix $\mathbf{S}$, and $\|\mathbf{a}\|=r$. Then $$ \mathbf{a}^T
\mathbf{S}\mathbf{a} gt; \frac{1}{2}\lambda_{\min} r^2 $$/p pWhat
properties of matrices were used to obtain the result? Thanks!/p