what is the physical meaning of diagonalization?

Answer 1

The image of the unit spehe |x|=1 under a diagonal matrix is an oval shaped figure. Unit sphere: x1^2+x2^2+...+xn^2 =1, if diagonal matrix maps x1->a1*x1, x2->a2*x2, etc. then the resultant vector w= diagonal matrix *x = (a1x1, a2x2, ..., anxn) solves (w1/a1)^2 +(w2/a2)^2 + ... +(wn/an)^2 = 1. In other words a diagonal matrix maps a point on the sphere (|x|=1) to a point on some oval shape (w1/a1)^2 +(w2/a2)^2 + ... +(wn/an)^2 = 1.

Diagonalization is the process of finding the correct bases (eigenvectors) so that you can think of the general matrix A as a stretching and shrinking (and maybe fliping) along the directions of the e-vectors (sphere to oval). Other than that it is just stretching space in those directions. (IF the matrix can be diagonalized!!!!) (IF you are patient enough to diagonaliz it --- hahaha. It is nice aka good work to work out one four by four by hand just to look see that it really really does work then use technology afterwards.)

Answer 2

Diagonalization:

The term diagonalization is used in two different senses in mathematics:

The process of finding a diagonal matrix similar to a given square matrix or representing a given linear map.
A certain proof technique used to show that one set is larger than another. Examples are Cantor's diagonal argument to show that the set of real numbers is uncountable, the proof of Gödel's incompleteness theorem, and Turing's proof that no algorithm can solve the halting problem.

Guido

Answer 3

making all the elements 0 exapt for the 1's on the daigonal