I am having some difficulty with Matrix multiplication properties, and I was wondering if someone could assist? Here is the problem:
Suppose there is an unknown Matrix $A\in\mathbb{R}^{2n\times 2n}$.
Also, there are two known Matrices $X\in\mathbb{R}^{n\times 2n}$ and $Y\in\mathbb{R}^{n\times 2n}$, both non-zero.
If we are given the following:
- $Z_1 = XAX^T$
- $Z_2 = XAY^T$
- $Z_3 = YAX^T$
Is it possible to construct $Z_4 = YAY^T$ from this information? If not, can you prove that it isn't possible, and what other information would be required?
Thank you.
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Example:
X = [ 0.5 0.5 0 0; 0 0 0.5 0.5]; Y = [-0.5 0.5 0 0; 0 0 -0.5 0.5];