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I am just curious about it because my examples for my final assignment involve many positive definite matrix. When I want to unitary diagonalize a positive definite matrix, its eigenvectors already orthonormal. So, is it true that eigenvectors of every positive definite matrix are orthonormal? Or are there some conditions for the matrix so that the eigenvectors can be orthonormal? Or is it just my example?

Note : what I know is every hermitian matrix has orthogonal eigenvectors (corresponding with distinct eigenvalues).

Sorry I can't show my examples because of "bad" numbers (float numbers). And sorry for my bad english.

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By definition, every positive definite matrix is symmetric. And eigenvectors corresponding to distinct eigenvalues of real symmetric matrices are always orthogonal.

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    $\begingroup$ That's right, but is it orthonormal? $\endgroup$
    – RANGGAJAYA CIPTAWAN
    Commented Dec 26, 2019 at 9:50
  • $\begingroup$ @RANGGAJAYACIPTAWAN Who cares if the eigenvectors are normalized or not? If you don't want the spectral decomposition, perhaps it does not matter. However, it seems that your question is motivated precisely by the spectral decomposition. $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 26, 2019 at 9:52
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    $\begingroup$ @RANGGAJAYACIPTAWAN That question doesn't make sense. If $v$ is an eigenvector of a matrix $A$, then any vector of the form $\lambda v$, with $\lambda\neq0$, is also an eigenvector of $A$. $\endgroup$
    – José Carlos Santos
    Commented Dec 26, 2019 at 9:59
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    $\begingroup$ Apparently, some authors apply "positive definite" even to non-symmetric matrices. $\endgroup$
    – Conifold
    Commented Dec 26, 2019 at 10:05
  • $\begingroup$ @Rodrigo de Azevedo You're right. I want to use spectral decomposition on a positive definite matrix. But every time I find the eigenvectors, its norm is already 1. So I don't need to normalize it to get a matrix unitary $U$. $\endgroup$
    – RANGGAJAYA CIPTAWAN
    Commented Dec 26, 2019 at 10:23

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