For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

A square matrix $A$ is diagonalisable if there is an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. One can view $P$ as a change of basis matrix so that, if $A$ is viewed as the standard matrix of a linear map $T$ from a vector space to itself in some basis, it is equivalent to say there exists an ordered basis such that the standard matrix of $T$ is diagonal. Diagonal matrices present the eigenvalues of the corresponding linear transformation along its diagonal. A square matrix that is not diagonalizable is called defective.

Not every matrix is diagonalisable over $\mathbb{R}$ (i.e. only allowing real matrices $P$). For example, $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

Diagonalization can be used to compute the powers of a matrix $A$ efficiently, provided the matrix is diagonalizable.

Diagonalization Procedure :

Let $A$ be the $n×n$ matrix that you want to diagonalize (if possible).

  • Find the characteristic polynomial $p(t)$ of $A$.
  • Find eigenvalues $λ$ of the matrix $A$ and their algebraic multiplicities from the characteristic polynomial $p(t)$.
  • For each eigenvalue $λ$ of $A$, find a basis of the eigenspace $E_λ$. If there is an eigenvalue $λ$ such that the geometric multiplicity of $λ$, $dim(E_λ)$, is less than the algebraic multiplicity of $λ$, then the matrix $A$ is not diagonalizable. If not, $A$ is diagonalizable, and proceed to the next step.

  • If we combine all basis vectors for all eigenspaces, we obtained $n$ linearly independent eigenvectors $v_1,v_2,…,v_n$.

  • Define the nonsingular matrix $$P=[v_1\quad v_2\quad …\quad v_n]$$
  • Define the diagonal matrix $D$, whose $(i,i)$-entry is the eigenvalue $λ$ such that the $i^{th}$ column vector $v_i$ is in the eigenspace $E_λ$.
  • Then the matrix A is diagonalized as $$P^{−1}AP=D$$

References:

Diagonal Matrix on Wikipedia

Matrix Diagonalization on Wolfram MathWorld