# 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?

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.

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).