Inner product of matrices and direction

Question

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.

Answer 1

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.