I am looking at the youtube lecture on determining positive semidefitiness by taking inner product of say (2x2) vector. The lecturer shows the following matrix: $$\begin{pmatrix} a & b \\ b & a \end{pmatrix}$$
but then he mentions if you compute inner product of vector $(x,y)$ with itself, we get the following expression:
$$ax^2+2bxy+by^2 >0$$
I would like to ask how did he jump from the above matrix to this expression? say the matrix above is $A$, taking inner product of itself do we have to do this? $AA^T$?
Not an answer, but too long for a comment.
If $A=\begin{bmatrix} a & b \\ b & a \end{bmatrix}$, then $A\begin{bmatrix} x \\ y \end{bmatrix}= \begin{bmatrix} ax+by \\ bx+ay \end{bmatrix}^T$ and $\begin{bmatrix} x \\ y \end{bmatrix}^T A \begin{bmatrix} x \\ y \end{bmatrix}= a x^2+2bxy + a y^2$.