# Are the eigenvectors of a positive definite matrix orthonormal?

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.

• @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 $A$. Commented Dec 26, 2019 at 9:59
• @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 $U$. Commented Dec 26, 2019 at 10:23