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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?

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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.

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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).

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