Diagonalizable

Definitions:

A linear operator T on a finite-dimensional vector space V is called $\textbf{diagonalizable}$ if there is an $\textbf{ordered basis}$ $\beta$ for V such that $[T]_\beta$ is a diagonal matrix.
A square matrix A is said to be $\textbf{diagonalizable}$ if A is similar t oa diagonal matrix. This $\beta$ is arbitrary. Basicaly the idea of diagonalizable is to use $\textbf{change of basis}$ . The key word is “if there is an ordered basis”. That is to say, if the $\textbf{linear representation}$ $[T]_\beta$ of the linear operator T can happen to be written as a diagonal matrix. Then we know this linear operator T is diagonalizable. However, this $\beta$ is not easy to find naturally. Therefore, We can write T as any other basis $\beta_2$ , and change it to a diagonal matrix $[T]_\beta$ by using formula for changing basis .

Any square matrix at least have an eigenvector. If it’s not in R, it’s in C.