# Magical generalizations thanks to Eigenvalues

Imagine taking ubiquitous scalar function $f(x)$, like $\exp(x)$ or $\sin (x)$, generalizing it somehow to a matrix function $f(\mathbf A)$, and expecting some of the properties of the scalar function to hold for the matrix function as well. Sounds like a magic trick, right? But generalizations of this kind are possible, and it’s precisely the point of this article to show some of them! This article also strongly supports the statement *that eigenvalues are like the chromosomes or genes of a matrix* (quasi-quote from Carl D. Meyer p543), as these magical generalizations require but the eigenvalues…

Imagine, say, defining matrix functions $\cos \mathbf A$ and $\sin \mathbf A$, and expecting one of the main properties of their scalar counterparts, namely $$\cos^2 x + \sin^2 x =1$$ to hold for the matrix functions as well: $$\cos^2 \mathbf A + \sin^2 \mathbf A =\mathbf 1.$$

How on earth can you even hope to define functions $\cos \mathbf A$ and $\sin\mathbf A$ that would satisfy this property? Clearly, a naive extension of the kind $[\cos (\mathbf A)]_{ij} = \cos([\mathbf A_{ij]})$ wouldn’t work, as you can easily try for yourself.

It turns that the key part to achieve the desired generalization is … the eigenvalues. Suppose that we have a diagonalizable matrix $\mathbf A$. We can take any scalar function $f(x)$ and define a straightforward matrix generalization $f(\mathbf A)$ as $$f(\mathbf A) = \mathbf P f(\mathbf D) \mathbf P^{-1} = \mathbf P \text{diag}(f(\lambda_1), f(\lambda_2), \dots, f(\lambda_n)) \mathbf P^{-1},$$ where $f(\lambda_i)$ is the scalar function of the $i$th eigenvalue $\lambda_i$.

If you use the extension above on some well-known functions, you can see for yourself that the key properties of a good number of standard scalar functions hold for the matrix function as well. One of the most insane examples is the infinite geometric series $$\sum\limits_{k=0}^\infty \mathbf A^k.$$ A well-known fact from algebra is that, the scalar geometric series $\sum\limits_{k=0}^\infty = x^k$ converges to $(1-s)^{-1}$ if $|x|<1$. How on earth would one expect this to hold for the matrix generalization as well? But it very much does, and under analogous assumptions! That is, if $||\mathbf A||<1$, then $$\sum\limits_{k=0}^\infty \mathbf A^k = (\mathbf{I-A})^{-1}.$$ If you you have a deeper understanding of math than I do, then you none of these may seem as a surprise to you… Still, this doesn’t mean that the generalization is not magical—it simply means that you are a magician yourself so you can see the “trick” behind the curtain.

If you didn’t find the generalizations above impressive, a few more perhaps will convince you about the strange power of the generalization of the eigenvalues:

- $\exp \mathbf 0 = \mathbf I$
- $\exp \mathbf A = \sum_{i=0}^{\infty} \frac{1}{i!} \mathbf A^i$
- $\exp^{\mathbf{A+B}} = \exp^{\mathbf A}\exp^{\mathbf B}$ (whenever $\mathbf{AB=BA}$)
- $(\sqrt{\mathbf A})^2 = \mathbf A$ (for nonnegative definite $\mathbf A$).

As you see, the generalization above took nothing but the eigenvalues. Makes one understand better why people have spent so much time to understand these magical numbers.