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I have the following question:

Matrix $N$ is a diagonal matrix with all entries strictly positive (hence, $N$ is positive definite since it satisfies $x^T N x > 0$). Matrix $M$ is an asymmetric positive definite matrix with all entries non-negative.

Since $NM \neq MN$, it does not follow that the product $NM$ is positive definite. However, given the special structure of $N$, can we still show that $NM$ is positive definite? Or maybe, under certain additional conditions?

  • 1
    What do you mean by positive definite for an asymmetric matrix? There are different options.2012-09-02
  • 0
    I meant that M also satisfies $x^T M x > 0$.2012-09-02

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