Re: Matrix inverse question


Hi,

Say I is the identity matrix, and M and K are real, symmetric positive
semi-definite (psd) matrices. All are n x n matrices, to be concrete.

Is I+MK necessarily invertible?

Yes.

If MK were PSD (e.g. if M and K had t same eigenvetors), then it's
obviously true, but MK is not even symmetric in general.

M has a psd square root, which I'll write as M^(1/2).
Now M^(1/2) K M^(1/2) is psd, and
MK = M^(1/2) (M^(1/2) K) and M^(1/2) K M^(1/2) = (M^(1/2) K) M^(1/2)
have the same eigenvalues (for square matrices, AB and BA have the
same characteristic polynomial).
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

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    ... Say I is the identity matrix, and M and K are real, symmetric positive ... If MK were PSD, ... We can extend this to the case M is semi-definite ...
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  • Matrix inverse question
    ... Say I is the identity matrix, and M and K are real, symmetric positive ... semi-definite matrices. ... If MK were PSD (e.g. if M and K had t same eigenvetors), ...
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