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Recall that 2 matrices $A, B\in R^{n,n}$ are similar if there exists a matrix $P$ such that $A=P^{-1}BP$.

In this case is $P$ always orthogonal?

dresden
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1 Answers1

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For $A$ and $B$ (over $\mathbb{R}$) to be similar, we only require that there be an invertible $P$ such that $A=P^{-1}BP$. The matrix $P$ clearly need not be orthogonal. This is the general definition of similarity for metrices.

We know also that every symmetric matrix is similar to a diagonal matrix and $P$ may be chosen to be orthogonal (but is not the only choice even in this case).

So if you wish you can require $P$ to be orthogonal, and then the matrices $A$ and $B$ are said to be orthogonally similar.

AnyAD
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