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?
Best Answer
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).