Suppose $A$ and $B$ are positive-definite and $x\in [0, 1]$. How do I show that eigenvalues of the following quantity are the same regardless of $x$?
$$ A^x B A^{1-x}$$
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Since $A$ is positive definite, $A^x$ is unambiguously defined on $\mathbb R$ using primary matrix function. Thus $A^{1-x}A^x=A$ and the result follows because $XY$ and $YX$ have the same spectra when $X$ and $Y$ are square matrices (so that the spectrum of $A^xBA^{1-x}$ is always equal to that of $AB$).
user1551
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how do I know that XY and YX have the same spectra? – Yaroslav Bulatov Nov 04 '19 at 18:53
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@YaroslavBulatov See the various answers of this question, for instances. – user1551 Nov 04 '19 at 19:19