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I am aware that the product of two symmetric positive definite matrices, $\mathbf{A}$ and $\mathbf{B}$, may not be a symmetric positive matrix. However, if both $\mathbf{A}$ and $\mathbf{B}$ are 2x2, then my own derivations indicate that one can always find a 2x2 rotation matrix $\mathbf{R}(v)$ with rotation angle $v\in (-\pi/2,\pi/2)$, such that $\mathbf{ABR}(v)$ is a symmetric positive definite matrix. Is this correct? If so, I also wonder whether this result can be generalized to nxn matrices?

Sketch of proof: I prove this using brute force for the 2x2 case. Each symmetric positive definite matrix can be described by three parameters. I multiply the three matrices and note that the resulting matrix $\mathbf{ABR}(v)$ is symmetric for a specific rotation angle $v^{*}\in (-\pi/2,\pi/2)$. My derivations actually indicate that symmetry of $\mathbf{A}$ and $\mathbf{B}$ is sufficient to get the weaker result that $v^{*}\in [-\pi/2,\pi/2]$. In the next step, I prove that the resulting symmetric matrix $\mathbf{ABR}(v^{*})$ is positive definite if $\mathbf{A}$ and $\mathbf{B}$ are both positive definite.

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    $\begingroup$ Please include your work on these derivations, if you don't mind. $\endgroup$ – The Count Feb 10 '17 at 16:05
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    $\begingroup$ It seems that your title question and the body of your question are different. The title seems to ask about any A, B, and R. The body seems to ask if there is at least on R that makes the relation true. $\endgroup$ – Brick Feb 10 '17 at 16:06
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    $\begingroup$ @user415090: You have basically discovered the Polar Decomposition. $\endgroup$ – Martin Argerami Feb 11 '17 at 3:21
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Let $A$ be any $n\times n$ matrix. Then $A=U\Sigma V^{*},$ for some unitary matrices $U$ and $V,$ and a diagonal matrix $\Sigma$ such that $\Sigma_{i,i}\geq0$ for all $1\leq i\leq n.$ This means that $A(VU^{*})=U\Sigma U^{*}$ is positive semidefinite, and $VU^{*}$ is clearly unitary. In general, $VU^{*}$ will not be a rotation matrix, however. In particular, this clearly holds if $A$ is of the form $A'B$ for some positive definite matrices $A'$ and $B.$

In the case $n=2,$ unitary matrices are quite restricted, and are essentially all rotation matrices (possibly with some rows or columns multiplied by $e^{i\theta}$ for some values of $\theta$), which explains your calculations in that case.

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