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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