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What are some lower bounds (if they exist) on $w^t A w$ in terms of $||w||_2^2$ for $A$ an matrix with all positive values? The lower bound can depend on $A$ and $\|w\|_2^2$. $w$ is $k \times 1$. Note when $A = I$ then we exactly get the squared norm of $w$.

If there are no such values, what in the case that $A$ is positive semi-definite? (I.e., $A$ has a $B$ such that $A = BB^T$ and then we are talking $w^tA w = \|Bw\|^2_2$ and we want it to be larger than something that depends on $\|w\|^2_2$).

Thanks.

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    @Jyotirmoy: The spectral norm gives an upper bound on $\|Aw\|$, which is not necessarily equivalent to the upper bound on $w^TAw$ unless $A$ is symmetric. Come to think of it, there's no point computing $w^TAw$ on a non-symmetric $A$, but still, it wasn't specified in the question...2010-10-14

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The bounds on $w^TAw$ are $\lambda_{\min} \|w\|_2^2 \le w^TAw \le \lambda_{\max}\|w\|_2^2,$ where $\lambda_{\min}$ and $\lambda_{\max}$ are the smallest and largest eigenvalues, respectively, of $\frac{1}{2}\left(A + A^T\right)$. Each bound is attained when $w$ is parallel to the corresponding eigenvector.