Does $\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$ hold for $A,B$ positive semi-definite matrices?

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Is

$$\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$$ where $I$ is identity matrix, $A,B$ are positive semi-definite complex valued matrices and $A^*$ is the conjugate (Hermitian) transpose of $A$.

Thanks a lot in advance. Question related to Possible matrix-determinant identity

We don't usually care whether something's homework or not. What's way more important is to show some self work, some ideas... — DonAntonio, Oct 06 '13 at 13:16

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

For complex matrices, positive-semi definite implies hermitian. So $A^*=A$.

Edit: see my answer here for a proof.

edited Apr 13 '17 at 12:21

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answered Oct 06 '13 at 13:16

Martin Argerami

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Does $\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$ hold for $A,B$ positive semi-definite matrices?

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