$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it where $S$ is a diagonal matrix. Also- does this similarity hold if $S$ was square but not-diagonal?
2026-04-07 11:21:56.1775560916
Similar matrix proof
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$S^{\frac{1}{2}}(S^{-1}T)S^{\frac{-1}{2}} = S^{\frac{-1}{2}}TS^{\frac{-1}{2}}$ as long as $S^{\frac{1}{2}}$ is uniquely specified (assuming it exists- if it didn't, the question would not be meaningful anyway). This does not require $S$ to be diagonal.