Matrix Multiplication in an Inequality Assume I have the following equation: $$ A_1 \vec x_1 \circ \vec y_1 \gt A_2 \vec x_2 \circ \vec y_2 $$
Here $ A_1 $ and $ A_2 $ are some $n \times n$ matrices and $ \vec x_1 , \vec x_2 , \vec y_1 , \vec y_2 $ are vectors of length $n$.
As such, the end result on either side is a constant. But what if I now want to multiply both sides on the left by $ A_1^{-1} $ (assuming A is an invertible matrix).
Can I do this?  Do I need to flip the sign? Does it depend on the determinant of $ A_1^{-1} $?
How do I deal with multiplying my matrices in an inequality?
 A: The answer is: you can not multiply by the inverse matrix $A_1^{-1}$. Just to see the inequality with another notation: consider $\vec x,\vec y\in\mathbb R^n$, then
$$\langle \vec x,\vec y\rangle=\vec x^{T}\vec y$$
where $\langle\cdot,\cdot\rangle$ denotes inner product in $\mathbb R^n$. Considering this, you can rewrite your inequality follows
$$\langle(A_1\vec x_1)^T,\vec y_1\rangle > \langle(A_2\vec x_2)^T,\vec y_2\rangle$$
As you said, each side is a real number, but you can see it makes no sense to multiply by a matrix in a case like this (exactly the same as you have written, but with a different notation).
A: As user159420 said, your question doesn't make sense as stated.  In general, one must be very careful with matrix multiplication and inequalities.  For example,
1) $x_1 > x_2 \not\Rightarrow  A_1x_1 > A_2x_2$.  Take for a counter example
$\begin{align}
x_1 &= \begin{pmatrix}1 \\ 1 \end{pmatrix} & 
x_2 &= \begin{pmatrix}-2 \\ -2 \end{pmatrix} &
A_1 &= \begin{bmatrix}2 & 1 \\ 1 & 1 \end{bmatrix} &
A_2 &= \begin{bmatrix}-3 & 1 \\ 1 & -3 \end{bmatrix} & 
\end{align}$
Then, $y_1 = A_1x_1 \not > A_2x_2 = y_2$.  Furthermore, along the lines of your question,
2) $ A_2x_2 > A_1x_1 \not\Rightarrow x_2 > A_2^{-1}A_1x_1$, as shown by the above example.
Good sources on inequalities with matrices generally show up in optimization theory, and specifically in linear and non-linear programming.  A wonderful source is Nonlinear Programming by Olvi L. Mangasarian.
