# Questions tagged [positive-semidefinite]

Relating to a symmetric $n\times n$ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$

432 questions
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### Inequality w.r.t determinant of a non-negative definite matrix.

I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks. For any non-negative definite (i.e. ...
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### Numerically verify a matrix is positive semidefinite

I am trying to numerically (in Julia) verify that A symmetric matrix $\mathbf{A}$ is positive semidefinite if and only if it is a covariance matrix. Then I need to verify in both directions, i.e. ...
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### Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
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### Positivity of a matrix

Let $A$ be a $3\times 3$ matrix defined in the following way: $$A=\begin{bmatrix} a & c & 0\\c & b &-c\\0 & -c & 1-a-b\end{bmatrix}$$ I wish to show that $A=BB^t$ for some ...
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### How to simplify this trace term?

I have the following trace term: trace(Sk' Ck Sk) where Sk is a KxM matrix and Ck is a KxK positive semidefinite matrix. I'm involving this trace term in a ...
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