Math for fun: Determining the angle between matrices $A$ and $B$ with three columns and rows

Question

So I was bored, as there were no math problems on Youtube that I thought that I might be able to solve, let alone any math videos, so I decided to start to do some determining the angle between two matrices using the Euclidean Inner Product, and got bored of that when I started to feel like it was a little too easy to determine the angle between two matrices with two columns and rows, so I asked myself, "What if I tried to determine the angle between two matrices $A$ and $B$ with three columns and rows? Surely I would be able to do that with the Euclidean Inner Product." And so, I created these two matrices:$$\begin{bmatrix}1&3&4\\3&4&5\\6&1&3\end{bmatrix}\text{ and }\begin{bmatrix}4&3&2\\3&6&4\\5&6&1\end{bmatrix}$$which I thought that I might be able to determine the angle between these two matrices. Here is my attempt at doing that:

According to the Euclidean Inner Product, the angle between two matrices is defined as$$\def\a#1#2#3#4{\left(\dfrac{#1_1#1_2+#2_1#2_2+\dots+#3_1#3_2+#4_1#4_2}{\sqrt{#1_1^2+#2_1^2+\dots+#3_1^2+#4_1^2}\sqrt{#1_2^2+#2_2^2+\dots+#3_2^2+#4_2^2}}\right)} \theta=\arccos\a {{(a_1)}}{{(a_2)}}{{(a_{n-1})}}{{(a_n)}}$$$$\therefore\arccos\left(\dfrac{A\cdot B}{|A||B|}\right)\text{ in this case }$$$$\equiv\arccos\left(\dfrac{4+9+8+9+24+20+30+6+3}{\sqrt{1+9+16+9+16+25+36+1+9}\sqrt{16+9+4+9+36+16+25+36+1}}\right)$$Which simplifies to$$\arccos\left(\dfrac{113}{\sqrt{122}\sqrt{152}}\right)$$$$=\arccos\left(\dfrac{113}{4\sqrt{61}\sqrt{19}}\right)$$$$\approx\arccos(0.829806312318)$$$$\approx0.592035810547^{\text{rad}}\text{ or }\approx33.921153265\unicode{xB0}$$

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My question

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Am I right in assuming that I would be able to use for any two matrices $A$ and $B$ assuming that $A$ and $B$ both have the same amount of rows and columns, or what would I use to determine the angle between the matrices?

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Mistakes I might have made

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1. Simplifying the Euclidean Inner Product
2. Using the Euclidean Inner Product
3. Using math symbols incorrectly

Answer 1

So firstly, you say you are determining the angle between two matrices but actually what you are doing is computing the angle between two vectors in $\mathbb{R}^9$ this is because depending on how you parameterize your matrix, the angles you get are different so really you are finding angles between points in $\mathbb{R}^9$. There is no natural identification in the sense that it works with matrix multiplication, this is mostly because there is no nice 'product' for points in $\mathbb{R}^k, k > 1$. Using your method you can compute the 'angle' between any two matrices that have the same number of elements, although this number really doesnt mean anything.

What you could try however if you really want to define an 'angle' between matrices you could try to define it on the spectrum of the product. Before going into detail there, an easy example to try is to define the angle between two complex numbers, this is done by this method. Once you understand that you can look at a commuting subalgebra of matrices and do something similar.