Positive Definite Matrices
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From: Manolis C. Tsakiris on 20 Apr 2008 02:29 Hello, i have two questions about positive definite matrices: 1)Let A>0, B Hermitian and AB<0. How can i prove that B<0? 2)Let A,B>0 and AB=BA. How can i prove that AB,BA>0? I verified the above statements via MATLAB simulation but i am investigating for a rigorous mathematical proof. any matrix analysis expert is welcome to try. Manolis
From: Andor on 20 Apr 2008 16:20 Manolis wrote: > Hello, > > i have two questions about positive definite matrices: > > 1)Let A>0, B Hermitian and AB<0. How can i prove that B<0? Hello Manolis! If B is Hermitian, it has a spectral decomposition (theorem). Take an eigenvector v of B and see what happens with v A B v. What does that tell you about the eigenvalues of B? > > 2)Let A,B>0 and AB=BA. How can i prove that AB,BA>0? If A and B commute they share a common eigenbasis (theorem). How then doe the spectral decompositions of AB and BA look like? Regards, Andor
From: Manolis C. Tsakiris on 20 Apr 2008 17:43
>Manolis wrote: >> Hello, >> >> i have two questions about positive definite matrices: >> >> 1)Let A>0, B Hermitian and AB<0. How can i prove that B<0? > >Hello Manolis! > >If B is Hermitian, it has a spectral decomposition (theorem). Take an >eigenvector v of B and see what happens with > >v A B v. > >What does that tell you about the eigenvalues of B? > >> >> 2)Let A,B>0 and AB=BA. How can i prove that AB,BA>0? > >If A and B commute they share a common eigenbasis (theorem). How then >doe the spectral decompositions of AB and BA look like? > >Regards, >Andor > ******************** Hello Andor, the common eigenbasis of A and B, providing they commute, was the missing key i was looking for! Be well Andor! |