Non-symmetric positive semi-definiteness in literature

Asked Mar 30 '23 at 01:12

Active Mar 30 '23 at 11:08

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The article Density matrix on Wikipedia includes the following sentence

Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.

As I was always taught positive semi-definite matrices are symmetric over real numbers and Hermitian over complex numbers, this seems curious to me. I know that matrices like

$$\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$$

always produce positive numbers over real vectors. But can you provide me any examples in literature or articles that use the broader definition of positive semi-definiteness?

edited Mar 30 '23 at 11:08

Rodrigo de Azevedo

asked Mar 30 '23 at 01:12

SlowerPhoton

1

some authors, but I have never seen it. In any case, adding a skew symmetric matrix to an existing matrix does not change the quadratic form. Add $\begin{pmatrix}0 & -\frac{1}{2} \ \frac{1}{2} & 0\end{pmatrix}$ to yours – Will Jagy Mar 30 '23 at 01:32
Quoting Qiaochu Yuan: "Positive-definite should not be a term that applies to matrices. It should only apply to quadratic forms, which are naturally described by symmetric matrices only" – Rodrigo de Azevedo Mar 30 '23 at 11:10
Related – Rodrigo de Azevedo Mar 30 '23 at 11:11

Non-symmetric positive semi-definiteness in literature

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