If $X$ and $W$ are real, square, symmetric, positive semidefinite matrices of the same dimension, does $XW + WX$ have to be positive semidefinite?
This is not homework.
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asked Oct 28, 2012 at 1:51
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3 Answers
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To answer the second part of your question, the matrix $XW+WX$ need not be positive semidefinite. Let $$X = \left(\begin{array}{rr} 4 & 2 \\ 2 & 1 \end{array} \right).$$ Let $$W = \left(\begin{array}{rr} 4 & -2 \\ -2 & 1 \end{array} \right).$$ Let $$v = \left(\begin{array}{r}0\\1\end{array}\right). $$
Then $$v^T XW v + v^T W X v = \big( \ 2 \ 1 \ \big) \left(\begin{array}{rc}-2 \\1\end{array}\right) + \big(\,-\!2 \ 1 \ \big) \left(\begin{array}{c} 2 \\1\end{array}\right) = -6 < 0. $$
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$\begingroup$ Prof. Shor: I am honored that you took the time to answer my humble question. $\endgroup$ Oct 28, 2012 at 14:40
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$\begingroup$ @navigetor23 : My question was $tr(XW) \geq 0$, and I also wondered if $XW + WX$ had to be positive semidefinite. I discovered the answer to my first question online after I posted. You posted a better reason why $tr(XW) \geq 0$, but someone else proved that $XW + WX$ need not be positive semidefinite and I accepted their answer. Unfortunately I can only accept one answer. $\endgroup$ Oct 28, 2012 at 16:54
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$\begingroup$ @navigetor23 : I should have split it up into two questions. I didn't because I was expecting $XW + WX$ to be positive semidefinite. $\endgroup$ Oct 28, 2012 at 17:02
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$\begingroup$ @navigetor23 : I edited my question so it asked if $XW + WX$ had to be positive semidefinite. I will re-ask why $XW$ has to have nonnegative trace and accept your answer if you give the same answer as before. $\endgroup$ Oct 28, 2012 at 17:08
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If $A, B$ are real, pos, and symmetric, then $A=A^{1/2}A^{1/2}$ and the trace of $AB$ is the trace of $A^{1/2}A^{1/2}B$ which is the the trace of $A^{1/2}BA^{1/2}$ which is a positive semidefinite matrix. Thus trace of $AB$ is nonnegative.
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$\begingroup$ How do you know $A^\frac{1}{2} A^\frac{1}{2}B$ and $A^\frac{1}{2} B A^\frac{1}{2}$ have the same trace, unless $A^\frac{1}{2}B$ is symmetric? $\endgroup$ Oct 28, 2012 at 15:07
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2$\begingroup$ The cyclic property of trace: $\mathrm{Tr} ABC$ = $\mathrm{Tr} BCA$. $\endgroup$ Oct 28, 2012 at 15:09
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$$XW \sim ZX\underbrace{Z^{-1}Z}_IWZ^{-1} = DZWZ^{-1}$$
Since $X$ is symmetric, square, and positive semi-definite, $D$ is diagonal for some $Z$ and has non-negative elements.
$ZWZ^{-1} \sim W$ has positive trace (trace is invariant to similarity and $W$ is positive square and semi-definite). There can be no negative values in the diagonal for $W$ and the following shows why.
Let $\mathbf{e}_i$ be the unit vector for the index $i$ that is the same index as the supposed negative diagonal value in $W$ (or any similarity to $W$). Then $\mathbf{e}_i^\top W \mathbf{e}_i = W_{ii} < 0$ which contradicts that $W$ is positive semidefinite.
