Test if a 2×2 numeric matrix is diagonalizable:

Test if a 3×3 symbolic matrix is diagonalizable:

Put the eigenvectors of the matrix in columns:

Verify the matrix is similar to diagonal matrix:

If a matrix is diagonalizable, it can be decomposed as , where is nonsingular and is diagonal. This can be used to compute a matrix function by using its power series representation as , and can be computed by simply applying to each of the diagonal elements. Use this method to compute several different matrix functions:

Compute the eigensystem of :

The matrix has the eigenvectors as columns, and the diagonal entries are the eigenvalues:

Compute and confirm using MatrixPower:

Compute and confirm using MatrixExp:

Compute and confirm using MatrixLog:

Compute and confirm using MatrixFunction:

Solve the system of ODEs , , . First, construct the coefficient matrix for the right-hand side:

The coefficient matrix is diagonalizable, so computing the matrix exponential will be straightforward:

Find the eigenvalues and eigenvectors:

Construct a diagonal matrix whose entries are the exponential of :

Construct the matrix whose columns are the corresponding eigenvectors:

The general solution is , for three arbitrary starting values:

Verify the solution using DSolveValue:

Use the fact that the recurrence relation for Fibonacci, , can be written in matrix form as to derive a formula for the pair . The solution to the recurrence relation is , where :

The matrix is diagonalizable:

Thus, the matrix power can be computed simply using Eigensystem[m]:

Complete the definition of :

Compute and :

Confirm the result using Fibonacci:

Normal matrices are the most general kind of matrix that can be unitarily diagonalized as with diagonal and unitary. All Hermitian matrices are normal because both sides of the equality are simply :

Similarly, all antihermitian matrices are normal because both sides of the equality are simply :

Unitary matrices are normal, as substituting in the definition gives an identity matrix on both sides:

Show that the following matrix is normal, then diagonalize it:

Confirm using NormalMatrixQ:

A normal matrix like can be unitarily diagonalized using Eigensystem:

The entries on the diagonal can be arbitrary complex numbers:

Normalizing the eigenvectors and putting them in columns gives a unitary matrix:

Confirm the diagonalization :

Skipping normalization of the eigenvectors suffices to show that is diagonalizable (though not unitarily):

In quantum mechanics, the energy operator is called the Hamiltonian , and a state with energy evolves according to the Schrödinger equation . An important assumption is that any state can be written as a sum of eigenstates. Show that this is the case for Hamiltonian for a spin-1 particle in constant magnetic field in the direction:

The matrix is diagonalizable, so its eigenvectors must form a basis for :

Computing the eigensystem, the energy levels are and :

Confirm that the eigenvectors are linearly independent and thus a basis:

Normalize the eigenvectors:

Decompose an arbitrary state as a sum of the eigenvectors:

Many matrix distributions produce matrices that are guaranteed to be diagonalizable, including CircularRealMatrixDistribution:

CircularSymplecticMatrixDistribution:

GaussianOrthogonalMatrixDistribution:

GaussianUnitaryMatrixDistribution :