I have a matrix $A=R\Lambda R^{-1},$ where $R$ is a positive definite upper triangular matrix and $\Lambda$ is a positive definite diagonal matrix. Is $A$ also positive definite? Thank you.
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Normally a positive definite matrix is Hermitian. How do you define a positive definite triangular matrix? – A.Γ. Nov 05 '16 at 21:01
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$R$ is a square matrix with all the eigenvalues positive. – Hongyi Xu Nov 05 '16 at 21:03
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If you do not care for the matrix being Hermitian then note that the eigenvalues of $A$ are the same as those of $\Lambda$, thus, positive. – A.Γ. Nov 05 '16 at 21:07
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What is your definition of positive definite matrix? Because in most cases, your matrix has to be symmetric (Hermitian) as A.G. had point out. – Jacky Chong Nov 05 '16 at 21:07
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Here I don't need $A$ to be symmetric. If $x^T A x > 0$ for all non-zero vectors $x$, $A$ is positive definite. – Hongyi Xu Nov 05 '16 at 21:11