## What is a Diagonalizable Matrix?

A diagonalizable matrix is any square matrix or linear map where it is possible to sum the eigenspaces to create a corresponding diagonal matrix.

An n matrix is diagonalizable if the sum of the eigenspace dimensions is equal to n.

A linear map of T : V → V is diagonalizable if the sum of eigenspace dimensions is equal to dim(V),

A matrix that is not diagonalizable is considered “defective.”

The point of this operation is to make it easier to scale data, since you can raise a diagonal matrix to any power simply by raising the diagonal entries to the same. In this case, the diagonal matrix’s determinant is simply the product of all the diagonal entries

### How to Diagonalize a Matrix?

A is the n×n matrix to diagonalize:

- Find the characteristic polynomial p(t) of A.
Find eigenvalues of the matrix A and their multiplicities from the characteristic polynomial p(t).

- For each eigenvalue of A, find the basis of the eigenspace Eλ.
- If there is an eigenvalue that gives the geometric multiplicity of λ, dim(Eλ) less than the algebraic multiplicity of λ, then the matrix A is not diagonalizable.
Combine all basis vectors for all eigenspaces to obtain the linearly independent eigenvectors v1,v2,…,vn.

- Define the nonsingular matrix:
- S=[v1v2…vn].
- Define the diagonal matrix D, where the (i,i)-entry is the eigenvalue, such that the i-th column vector vi is in the eigenspace Eλ.

Then the matrix A would finally be diagonalized as:

S−1AS=D.