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.
Let us take an example of a vector V = i + 2j which in matrix notation is 



It will be represented as



POSITIVE DEFINITE MATRIX


A positive definite matrix is 



How do we know it is positive definite matrix



The altered vector is






POSITIVE SEMI-DEFINITE MATRIX




The altered vector is [0 , 0]


NEGATIVE MATRIX




The altered vector in this case is



We can see the quadrant has changed