Inner product of matrices and direction

In vector $$v\in \mathbb{R}^n$$, we can use inner product to say something

1. if $$u, v\in \mathbb{R}^n$$ are in the same direction, $$\left\langle \frac{v}{\|v\|_2}, \frac{u}{\|u\|_2}\right\rangle = 1$$
2. if $$\left\langle v,u \right\rangle>0$$, then we know the angle between them are less then $$90^\circ$$.

Can we define such concept in matrix? if we view a square matrix $$X\in \mathbb{R}^{n\times n}$$ as a vector, can we do the similar thing as above?

We know we have frobenius norm (for matrix) is the Euclidean norm of the vector by listing the coefficient of matricies.

• Can you be a bit more specific? Just mapping $X \in \mathbb{R}^{n\times n}$ into a vector $\mathcal{X} \in \mathbb{R}^{n^{2}}$ means we can directly use the results but do you mean something relating the row and column vectors of $X,Y \in \mathbb{R}^{n \times n}$? Commented Jan 27, 2021 at 11:42
• @AlphaNumeric 1. This is what I want to ask. If we have such property, i.e., definition of frobenius norm in matrix and its relation to vector, do we have similar concepts in matrices like these in vectors? 2. For the definition of frobenius norm, I think we can pile columns of a matrix up to form a vector. Commented Jan 27, 2021 at 12:24
• Space of matrices are just specific finite-dimensional vector spaces, and so plenty of inner product can be defined on them.
– fcz
Commented Jan 27, 2021 at 18:54

Yes, most certainly. Just like the degree of parallelity of two vectors is measured by the inner product of their norms,

$$\langle \widehat{\pmb{u}}, \widehat{\pmb{v}} \rangle$$,

The parallelity of two $$n \times n$$ matrices $$A$$ and $$B$$ is measured by their anticommutator,

$$\{ A, B \} = AB+BA$$

Similarly, the matrix operation which measures degree of orthogonality, corresponding to $$\widehat{\pmb{u}} \times \widehat{\pmb{v}}$$ in the case of vectors, is given by their commutator,

$$\left[ A, B \right] = AB-BA$$

This isn't surprising, considering the fact that the notion of cross products can be generalized using the wedge product, defined as,

$$\pmb{u} \wedge \pmb{v} = \pmb{u} \otimes \pmb{v} - \pmb{v} \otimes \pmb{u}$$

Where $$\pmb{u} \otimes \pmb{v} = \pmb{u} \pmb{v}^T$$

These are just general tricks which when applied to matrices, tell us about the 'angle' between them in the matrix space $$\mathbb{R}^n \mapsto \mathbb{R}^n$$

Note: Just as in the case of vectors, we consider their norms to find the pure angle between them; in the case of matrices $$A$$ and $$B$$, we really apply the above operations to $$\frac{A}{\det \left( A \right)}$$ and $$\frac{B}{\det \left( B \right)}$$ respectively.