How is this animation, deforming one linear transformation into another, parametrized?

Question

I just want to know what is the most 'sensible' way to parametrise a linear transformation, if there is such a way. For example in:

https://shadanan.github.io/MatVis/

Each linear map is parametrised so that when t = 0 it is the identity map, and when t = 1 it is the map itself. I just don't know where the maps in between t = 0 and t = 1 come from. It must be some kind of decomposition related to eigenvalues.

Answer 1

Let $I$ be the identity, and let $M$ be the $2\times 2$ matrix displayed in the upper left corner of the animation. The matrix-valued map

$$f(t)=I+(M-I)t$$

satisfies $f(0)=I$ and $f(1)=M$.

This is a standard trick: given a real vector space $V$ and "initial" and "terminal" vectors $v$ and $w$, the $V$-valued curve $f(t)=v+(w-v)t$ deforms $v$ into $w$ over the interval $t\in[0,1]$.