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 some operation on vectors.

The definition is:

A n × n Hermitian matrix M is called positive-semidefinite if

x^{*} M x \geq 0

for all x \in \mathbb{C}^n (or, all x \in \mathbb{R}^n for the real matrix), where x^* is the conjugate transpose of x.


One intuitive definition is as follows. Multiply any vector with a positive semi-definite matrix. The angle between the original vector and the resultant vector will always be less than or equal \frac{\pi}{2}. The positive definite matrix tries to keep the vector within a certain half space containing the vector. This is analogous to what a positive number does to a real variable. Multiply it and it only stretches or contracts the number but never reflects it about the origin.

Source : Link , Question Author : Community , Answer Author : Andreas G.

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