Pseudo diagonalization

In this video, I show that you can turn *any* square matrix into a diagonal matrix if you allow for different bases for the input and output spaces. This is not quite diagonalization, but pseudo-diagonalization, because diagonalization requires to have the same basis for both spaces. Still, it's a neat exercise in linear algebra.

[MsSlash]

This is the only video that doesn't start by saying "All right; thanks for watching!"

[ajokaefi]

Of course it works when the dimensions of V and W are not equal ... anyway, love your enthusiasm and knowledge!

[chirayu_jain]

I have a question for you
Prove that your subscribers diverge

[rigorless]

great video, i think i was able to follow it. now for a rant:


I think i would actually really enjoy linear algebra, but i just abhor the existing notations (and it doesn't help there's so many!). The way everything can be boiled down to matrix theory is nice, but all the ellipses (...), and the arrows pointing everywhere, and the fact that its hard to tell a variable/number apart from a vector, and the way parentheses are used (sometimes it means an operation on a vector, sometimes it means an operation on a space)


and why "m x n" for rows by columns, rather than "r x c" or something? (i always forget which is m and which is n) Why do we have ij notation to refer to individual elements in a matrix, but not individual elements in a vector? why is it okay for a row vector to be multiplied by a column vector *OR* another row vector? (and get the same answer) isnt a row vector basically just a matrix with 1 row and n columns? but for matricies, (1 x n) * (1 x n) is undefined? so why is it allowed for rows if rows are just a special case of a matrix? if rows are the exception, shouldnt we be defining row vectors first and then constructing the definition of a matrix using row vectors?


i feel like literally all my complaints about LA at some level boil down to the notations and starting definitions, which really shouldnt matter, and it would be such an elegant subject otherwise, but a lot of times it just comes off as messy.