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. $Ax$)?
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. $Ax$)?
If the matrix is not symmetric, there is no notion of positivity other than having eigenvalues all positive. For symmetric matrices, you get a partial ordering such as $A\succ B$ which means $A-B$ is positive definite.
Moreover, every symmetric positive definite matrix defines an ellipsoid. The principal axes are given by the eigenvectors of $A$ define and the square root of the eigenvalues are the radii of the corresponding axes. The usual convention is to use the inverse of the matrix to define the ellipsoid. This is to define the unit ball as the image of the mapping $x^TA^{-1}x$.
welcome to math.stackexchange. Assuming that you mean that $A$ is positive definite, one intuition is a parabola: From the definition of positive definite, we get that $x^\top A x > 0$ for all $x\not=0$. For example, if $A=\left(\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right)$ then $x^\top A x = x_1^2 + x_2^2$.
For a general p.d. $A$ you can first transform $x$ into the basis spanned by the eigenvectors of $A$. In that space $A$ becomes diagonal with the positive eigenvalues on the diagonal (positive since $A$ is p.d.). Then, the same intuition holds again.