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Let $A$ and $B$ both be positive definite matrices. How do I show that their Kronecker product is also positive definite?

I know we can use the fact that the eigenvalues of the Kronecker product is $\lambda_A+\lambda_B$ which are all positive. But I want to use a different approach here.

Thank you.

  • 0
    Use the definitions directly! The only "tool" you should need is a single application of the spectral decomposition of *one* of the matrices ($A$ or $B$, your choice!).2012-10-16
  • 0
    Are A and B the same dimensions?2012-10-16
  • 1
    @Bitwise: They don't have to be, no.2012-10-16
  • 4
    BTW I believe the new eigenvectors are the product of the pairs of A and B eigenvectors, not the sum. It still produces positive eigenvectors, though.2012-10-16
  • 0
    Use mix product property and diagonaliztion of $A$ and $B$.2012-10-16

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