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Let $M$ be a complex $n\times n$ matrix and $I$ the $n\times n$ identity matrix. The overline $\bar{M}$ denotes the complex conjugate. The determinant of $I+M\bar{M}$ is obviously real, because $\det(I+M\bar{M})=\det(I+\bar{M}M)=\overline{\det(I+M\bar{M})}$.
I have no doubt that $$\det(I+M\bar{M})\geq 0,$$ but I do not have a proof. Can someone help me?

Motivation: I need this to complete an answer at MO.

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  • $\begingroup$ MMbar is definite nonnegative, hence has no negative eigenvalues. Hence every eigenvalue of Id +MMbar is greater than one, hence the det (which is the product of these eigenvalues) is greater than one. $\endgroup$
    – G. Fougeron
    Feb 25, 2021 at 19:48
  • $\begingroup$ @G.Fougeron This is wrong. Note that $M$ is not assumed symmetric so $M \overline{M}$ is in general not hermitian. $\endgroup$
    – Robert Israel
    Feb 25, 2021 at 20:01
  • $\begingroup$ $M\overline{M}$ doesn't need to be nonnegative definite because $\overline{M}$ is the termwise complex conjugate and not the Hermitian transpose (I had to click through to the MO post to be sure of this). e.g. with $M = \begin{pmatrix} 0 & -2i \\ i & 0\end{pmatrix}$ you get $-2I$ $\endgroup$
    – leslie townes
    Feb 25, 2021 at 20:02
  • $\begingroup$ Ooops sorry, my bad, I though that there was a complex conjugacy thrown in. $\endgroup$
    – G. Fougeron
    Feb 25, 2021 at 23:18
  • $\begingroup$ math.stackexchange.com/questions/3477987 $\endgroup$
    – user26857
    Mar 3, 2021 at 8:46

1 Answer 1

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the 2 by 2 case with all letters real. We think the determinant is positive, attempt to write it as a sum of squares, I guess.

parisize = 4000000, primelimit = 500000
? m = [ a+b*I, c+d*I; e+f*I, g+h*I]
%1 = 
[a + I*b c + I*d]

[e + I*f g + I*h]

? n = [ a-b*I, c-d*I; e-f*I, g-h*I]
%2 = 
[a - I*b c - I*d]

[e - I*f g - I*h]

? matt = matid(2) + m*n
%5 = 
[a^2 + (b^2 + ((e - I*f)*c + ((I*e + f)*d + 1))) (c - I*d)*a + ((I*c + d)*b + ((g - I*h)*c + (I*g + h)*d))]

[(e + I*f)*a + ((-I*e + f)*b + ((g + I*h)*e + (-I*g + h)*f)) (e + I*f)*c + ((-I*e + f)*d + (g^2 + (h^2 + 1)))]

? dett = matdet(matt)
%6 = (g^2 + (h^2 + 1))*a^2 + 
((-2*g*e - 2*h*f)*c + (-2*h*e + 2*g*f)*d)*a +
 ((g^2 + (h^2 + 1))*b^2 +
 ((2*h*e - 2*g*f)*c + (-2*g*e - 2*h*f)*d)*b +
 ((e^2 + f^2)*c^2 + 2*e*c + ((e^2 + f^2)*d^2 + 2*f*d + (g^2 + (h^2 + 1)))))
?
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