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Question:

Let $A$ and $B$ be $n$ x $n$ matrices such that both A and B are orthogonally diagonalizable. Prove or disprove the following assertions:

(i) the matrix $AB$ is orthogonally diagonalizable;

(ii) the matrix $AB+BA$ is orthogonally diagonalizable.

I think I'll be using the property that $A=MTM^{-1}=MTM^T$ and same for $B$, but not really sure where to go from there? TIA.

Dawa
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  • with orthoganally diagonalizable you mean that are ortogonal and diagonalizable matrix? In this case, the eigenvalues are only $1$ and $-1$ (in $\mathbb{R}$) – Martín Vacas Vignolo Nov 04 '21 at 20:18
  • you wrote $A=M T M^T$ where $T$ is diagonal. It necessarily has real eigenvalues so $A$ is symmetric. Take a look at https://math.stackexchange.com/questions/3621512/a-and-b-are-real-symmetric-and-positive-semi-definite-matrices-of-the-same-orde/ then conclude you should come up with a counter example where $A$ is PSD but $B$ is indefinite (i.e. has positive and negative eigenvalues). Focus on the $2\times 2$ case for your counterexample. – user8675309 Nov 04 '21 at 20:56

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