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If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$

I am trying to prove this as follows:

If $v$ is an eigenvector of $A$, then $Av = \lambda v$ and we can rewrite the expression as follows: \begin{align*} v^\top A v = v^\top \lambda v= \lambda \left(v^\top v\right) \end{align*} As $\lambda > 0$ and $\left(v^\top v\right) = \vert\vert v\vert \vert_2 > 0$ for all $v \ne 0$. Therefore, $v A v > 0$ has to be true.

What if $v$ is not an eigenvector of $A$? is it possible? I know this should be a proof for symmetric matrices to be positive-definite, but how can I strengthen this proof?

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  • $\begingroup$ No the condition $Av = \lambda v$ is for eigenvectors. Your proof looks fine to me. $\endgroup$ – IAmNoOne Jul 6 '15 at 4:45
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    $\begingroup$ Hint - A symmetric real (thanks @hardmath) matrix $A$ is diagonalizable over the reals. Furthermore, all of the eigenspaces (corresponding to distinct eigen values) are mutually orthogonal. $\endgroup$ – peter a g Jul 6 '15 at 4:48
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    $\begingroup$ It might be useful to clarify that $A$ is real symmetric. $\endgroup$ – hardmath Jul 6 '15 at 4:50
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    $\begingroup$ $vAv$ ? I think it should be $v^TAv$ $\endgroup$ – Chiranjeev_Kumar Jul 6 '15 at 4:50
  • $\begingroup$ @Chiranjeev: Yes, thanks for pointing this out. $\endgroup$ – amaatouq Jul 6 '15 at 4:52
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By Spectral Theorem, $A$ is orthogonally similar to a diagonal matrix, i.e $$ P^{T}AP=\pmatrix{\mu_1 \\ & \ddots \\ && \mu_n} $$ where $\mu_i>0$ is eigenvalue of $A$, and $\space P^{T}P=P^{-1}P=I$.

For any $v$, let $v=Pu$. Then $$ v^TAv=u^TP^TAPu=\sum_{k=1}^n\mu_ku_k^2>0 $$

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  • $\begingroup$ Yup! That's how to show this one. +1 $\endgroup$ – Mark Viola Jul 6 '15 at 5:38
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If $A$ is a real symmetric matrix, then you can form an orthonormal basis for $\mathbb{R}^n$ from its eigenvectors. What happens when you expand $v$ in this basis?

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@hermes already provided the way forward. I thought that it might be instructive to see the use of tensor notation herein. So, here we go ...

Let $U$ be the orthogonal matrix that diagonalizes $A$, such that $UU^T=U^TU=I$ and $(U^TAU)_{ij}=\lambda_i\delta_{ij}$, where $\lambda_i$ is the $i$'th eigenvalue of $A$ and $\delta_{ij}$ is the Kronecker Delta.

Using tensor notation, with summation implicit over repeated indices, we have

$$\begin{align} (v^TAv)_{in}&=v_iU_{ij}U^T_{jk}A_{k\ell}U_{\ell m}U^T_{mn}v_n\\\\ &=v_iU_{ij}\lambda_j\delta_{jm}U^T_{mn}v_n\\\\ &=v_iU_{im}\lambda_mU^T_{mn}v_n\\\\ &=\lambda_m\left(v_iU_{im}\right)\left(v_nU_{nm}\right)\\\\ &=\sum_{m}\lambda_m\left(v_iU_{im}\right)^2\\\\ &>0 \end{align}$$

as expected!

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