Positive Definite

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A Positive Definite matrix, in a formal definition, is to define a Symmetric Matrix $A$ that satisfies the following condition:

A matrix $A$ is said to be positive definite if for any non-zero vector $z$, if

$$z^{\prime }Az>0\text{for all}z\ne 0.$$

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It may be too abstract to understand this concept, but thinking in single variable case:

Consider $a{x}^2$, if $a>0$, then for any $x\ne 0$, $a{x}^2>0$, so the matrix $[a]$ is positive definite. It ensures the positivity.