TITLE: what is the geometrical interpretation to positive definite matrix
QUESTION [10 upvotes]: What is the geometrical interpretation of positive definite matrix ?
(not necessarily symmetric)
if $A$ is positive definite, what does it do to a vector $x$ (i.e. $Ax$)?

REPLY [6 votes]: Matrix being positive definite is like a (real) number being positive.
Say you have a positive (real) number $a$. It has a property such that you can multiply any real number $b \in R$ by $a$ and the result of multiplication $a \cdot b$ will preserve $b$'s sign. In another words, $a$ scales $b$ but does not reflect it (or change its sign).
Now let's apply this concept to matrices.
First, recall that for any vectors $a, b$:
$$
a^T b = |a||b|cos\theta
$$
where $\theta$ is the angle between $a$ and $b$. The value of $a^T b$ is positive iff $cos\theta > 0$ (i.e. $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$)
Second, recall the definition of a positive-definite matrix $A \in R^{n \times n}$:
$$
x^T A x > 0
$$
for all $x \in R^n \backslash \{ 0 \}$.
The meaning of $x^T A x$ is: Take any vector $x$ and transform it into $Ax$ and then take a dot product of $x$ with its "transformed self". Since the result is always positive this must mean that the angle between $x$ and $Ax$ must have been such that $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$. This means that $A$ does not "reflect" any vector, thus behaving like a real (positive) number $a$.