Finding Cos of an angle between matrices

Question

I have two $2\times 2$ matrices and it's asking me to find the Cos of the angle between them. Firstly, how do yall visulize matrices and the angles between them? I think that's my first problem. Secondly, how can I visualize the inner product between 2 matrices? And lastly, I learned that the dot product is the standard inner product, how can I intuitively understand the inner product defined by $<A,B> = \mathrm{tr}(B^T A)$? That is, defined by transpose of $B$ multiplied by $A$?

Answer 1

If $\alpha$ is the angle between the matrices then you are asked to compute

$$\cos \alpha=\displaystyle \frac{\langle A,B\rangle}{\sqrt{\langle A,A\rangle}\sqrt{\langle B,B\rangle}}=\frac{\mathrm{tr}(B^TA)}{\sqrt{\mathrm{tr}(A^TA)}\sqrt{\mathrm{tr}(B^TB)}}.$$

This vectors (matrices) live in a four-dimensional space, which makes difficult to visualize some geometric quantities. But with that inner product you can identify your space with $\mathbb{R}^4$ with the following orthonomal basis $e_1=(1,0,0,0),e_2=(0,1,0,0),e_3=(0,0,1,0),e_4=(0,0,0,1)$ (that is, every vector has length one and any two of them are perpendicular).

The mentioned identification is as follows:

$$\left(\matrix{1 \: 0\\ 0 \: 0}\right)\rightarrow e_1, \left(\matrix{0 \: 1\\ 0 \: 0}\right)\rightarrow e_2, \left(\matrix{0 \: 0\\ 1 \: 0}\right)\rightarrow e_3, \left(\matrix{0 \: 0\\ 0 \: 1}\right)\rightarrow e_4 $$