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I am trying to prove that two determinants are equal. I've come quite far, but I do not know how to proceed and I cannot find any relevant properties. They contain the same matrices, but in a different ordering. Let $V$ be a $n\times n$ matrix, $H$ a $p\times n$ matrix, $H'$ its transpose, and $B$ a $p\times p$matrix:

$$\det(VH'BH+I_n)=\det(BHVH'+I_p)$$

Good references are also very welcome! I am currently mostly using the matrix cookbook.

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    $\begingroup$ What does $H'$ indicate? Is it the inverse? Are (some of) these matrices invertible? $\endgroup$
    – Andijvie
    Jun 22 at 8:14
  • $\begingroup$ H' indicates the transpose of H $\endgroup$
    – Kurt Z.
    Jun 22 at 8:16

1 Answer 1

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Let $C=VH'$ and $D=BH.$ You want to prove $$\det(CD+I_n)=\det(DC+I_p).$$

This is the well-known Sylvester determinant identity.

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