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Given symmetric positive definite (SPD) matrices $\bf A$ and $\bf B$, define the following matrix equation in ${\bf X} \in \mathrm{O}(n)$.

$$ {\bf X} {\bf A} {\bf X}^\top = {\bf B},$$

Given any SPD matrices ${\bf A}, {\bf B}$, does there exist an orthogonal solution ${\bf X}$? If ${\bf X} \in \mathrm{O}(n)$ exists, does it have a closed form expression? When ${\bf X}$ is confined within SPD matrices, I know this is a Riccati equation. So how about when ${\bf X} \in \mathrm{O}(n)$?

Rodrigo de Azevedo
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gsoldier
  • 175

1 Answers1

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Let $A$ and $B$ be symmetric real $n\times n$ matrices. The following statements are equivalent:

  1. There exists $X\in\mathrm{GL}(n,\mathbb{R})$ such that $XAX^{-1}=B$.
  2. There exists $X\in\mathrm{O}(n,\mathbb{R})$ such that $XAX^{T}=XAX^{-1}=B$.
  3. $A$ and $B$ have the same eigenvalues counting multiplicities.
  4. $A$ and $B$ have the same characteristic polynomial.

This follows from the spectral theorem.

Brauer Suzuki
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