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…
How to compute the “real” rank of a matrix?
If you fill an $n\times n$ matrix with random entries, than you’ll almost surely end up with a full-rank matrix. Also, any matrix that is constructed with real and continuous data (e.g., sensor input) will also be almost surely of full rank even if the underlying should have lead to linearly dependent columns/rows. Further, if we do not use exact arithmetic but, say, floating point arithmetic, our $\mathbf A$ will almost surely be somewhat perturbed, especially if it is a result…