When is a symmetric matrix positive semidefinite

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. It's actually defined more generally for Hermitian matrices matrices equal to the complex conjugate of their transpose. If a Hermitian matrix has at least one entry with a nonzero imaginary part, then the matrix is not symmetric, but it could still be positive semi- definite.

There are two reasons I can think of why we wouldn't extend the definitions to non-Hermitian matrices. The first is that if the matrix is not Hermitian, then we could have complex eigenvalues.

In that context, "positive" has no meaning. Another reason why I think we wouldn't has to do with quadratic forms. Unfortunately, not everybody does it this way, which can produce some confusion. And in the case of complex matrices, what you want is Hermitian, not symmetric. Basically the answer is that with the symmetry assumption, positive semi -definite matrices have some very nice properties, and without it they don't. A simple intuition is that positive definite matrices are matrices which eigenvalues are all strictly greater than zero.

The condition can be studied outside of bilinear forms, but in that case it is particularly useful and natural. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why is positive semi- definite only defined for symmetric matrices?

Ask Question. Asked 6 years, 10 months ago. Learn more. Ask Question. Asked 6 years, 6 months ago. Active 10 months ago. Viewed 1k times. Are symmetric matrices always at least positive semi-definite? Thanks so much, Jhance. Add a comment. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.

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