Diagonalizable Matrix

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:

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

  3. For each eigenvalue of A, find the basis of the eigenspace Eλ.
  4. 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.
  5. Combine all basis vectors for all eigenspaces to obtain the linearly independent eigenvectors v1,v2,…,vn.

  6. Define the nonsingular matrix:
  7. S=[v1v2…vn].
  8. 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.