Positive semidefinite inner product

If S,Z are positive semidefinite. We now know that $y^TSy=\sum_{i}\sum_{j}S_{ij}y_{i}y_{j}\geq 0$, same goes for Z. We also know that we can write S as $S=\sum_{i}\lambda_{i}x_{i}x_{i}^{T}$, with $\lambda_{i}$ the eigenvalues and $x_{i}$ eigenvectors of the matrix S. In other words $S_{ij}=\sum_{k}\lambda_{k}x_{ik}x_{jk}$ by construction.

I want to show that $$\sum_{i}\sum_{j}S_{ij}Z_{ij}\geq0$$ Could anyone help me show this??

I would also like to know why the equality of this statement only holds if and only if $S \cdot Z=0$

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Let $x_1, \dotsc, x_n$ be an orthonormal basis of eigenvectors for $Z$ with eigenvalues $\lambda_1, \dotsc, \lambda_n \geq 0$. Then

$$\sum_{i, j}S_{ij}Z_{ij} = \operatorname{Trace}(SZ) = \sum_i\langle SZ x_i, x_i\rangle = \sum_i \lambda_i \langle Sx_i,x_i \rangle \geq 0.$$

In case of equality each term of the summand must be zero. So either $\lambda_i = 0$ and $Zx_i = 0$ or $\langle Sx_i, x_i \rangle = \langle S^{1/2}x_i, S^{1/2}x_i \rangle = 0$ and $Sx_i = 0$. This shows that $SZx_i = 0$ for all indices $i$ and so $SZ = 0$.

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Okay, but how can I do it using $S=\sum_{i}\lambda_{i}x_{i}x_{i}^T$ ? I would prefer to not use the trace as I am not very familiar with it.. – Math Girl Nov 11 '12 at 14:53
The trace is just the sum of the diagonal elements (and also the sum of the eigenvalues). – John Moeller Jan 22 '13 at 22:28

Note $S_i=\lambda_i x_i x^\top_i$, and so, $S=\sum S_i$. To show that the dot product between $Z$ and $S$ satisfies $Z\cdot S\geq 0$, it is enough to show that $Z\cdot S_i\geq 0$. For this, it is enough to show that $\lambda_i Z\cdot x_i x^\top_i\geq 0$. Since $S$ is positive semidefinite, $\lambda_i\geq 0$. We only need to prove $Z\cdot x_i x^\top_i\geq 0$, which is simply equivalent to $x_i^\top Z x_i\geq 0$, which is true for any $x$ because Z is positive semidefinite.

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