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I have the square matrix equation (all matrices of dimension $n \times n$):

$G \Lambda G' = \Sigma$ where $\Lambda$ and $\Sigma$ are diagonal (with positive values on the diagonal). Under what sufficient conditions are the only solutions for $G$ equal

$G = \Sigma^{1/2} V \Lambda^{-1/2}$ for a $V$ such that $VV' = I$?

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    thanks, I didn't know about it.2012-07-20

1 Answers 1

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Multiply by $\Sigma^{-1/2}$ from left and right. The result can be written $VV'=I$ with $V=\Sigma^{-1/2}G\Lambda^{1/2}$, which yields your equation when solved for $G$, so there's no need for further conditions.