Test if a 2×2 numeric matrix is normal:

Test if a 3×3 symbolic matrix is normal:

Verify by hand the condition m.ConjugateTranspose[m]ConjugateTranspose[m].m:

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 EigenvalueDecomposition:

The entries on the diagonal can be arbitrary complex numbers:

While this is a diagonalization, it is not unitary:

The columns of are orthogonal, as they must be, but they are not normalized:

Normalizing the columns of gives a unitary matrix:

Confirm the unitary diagonalization :

Generate a normal matrix from a given set of eigenvalues using the fact that any normal matrix can be represented in the form with unitary and diagonal:

In general, the matrix is not Hermitian, antihermitian or unitary:

If the eigenvalues used to generate the matrix are real, the matrix is Hermitian:

If the eigenvalues are pure imaginary, the matrix is antihermitian:

If the eigenvalues lie on the unit circle, the matrix is unitary:

Create a matrix that is both unitary and antihermitian:

Because normal matrices are unitarily diagonalizable, computing matrix functions is particularly simple. If , then , and can be computed by simply applying to each of the diagonal elements. Consider the following normal matrix:

Compute Eigensystem[n]:

The eigenvectors are normalized in this case, so and is the diagonal matrix built from :

Compute and confirm using MatrixPower:

Compute and confirm using MatrixExp:

Compute and confirm using MatrixLog:

Compute and confirm using MatrixFunction:

Many matrix distributions produce normal matrices, including CircularRealMatrixDistribution:

CircularSymplecticMatrixDistribution:

GaussianOrthogonalMatrixDistribution:

GaussianUnitaryMatrixDistribution :

SchurDecomposition for complex normal matrices is always equivalent to Eigensystem:

The triangular matrix t is effectively diagonal, although it may contain roundoff errors:

The off-diagonal elements of t are roundoff errors:

The eigenvalues on the diagonal of t are not sorted:

NormalMatrixQ[m] is effectively equivalent to m.m^m^.m:

Any real symmetric matrix is normal:

As is any Hermitian matrix:

Any real antisymmetric matrix is normal:

As is any antihermitian matrix:

Any real orthogonal matrix is normal:

As is any unitary matrix:

For a normal matrix , for all vectors :

The product of two normal matrices need not be normal:

The product will be normal if and only if the two normal matrices commute:

In this case, the sum will be normal as well:

A matrix is normal if and only if its Hermitian and antihermitian parts commute:

A normal matrix is always diagonalizable:

Indeed, it is always unitarily diagonalizable using Eigensystem:

The eigenvectors are orthogonal:

Thus, placing the normalized eigenvectors in columns forms a unitary matrix:

Thus, , with the diagonal matrix whose entries are the eigenvalues :

Consider {s,j}=JordanDecomposition[m] for a normal matrix m:

The matrix is always diagonal:

The matrix has columns that are orthogonal but not necessarily orthonormal:

This can be verified more quickly using UnitaryMatrixQ with the setting NormalizedFalse:

Schur decomposition {q,r} and eigendecomposition {t,d} essentially coincide for a normal matrix:

Compute SchurDecomposition[n,RealBlockDiagonalFormFalse]:

The option ensures the right triangular matrix r will be diagonal:

equals up to ordering, and equals up to ordering and phase:

For this particular matrix, the ordering is the same:

To verify the equality of and , set the first entry of each column to 1. to eliminate phase differences:

A normal matrix with real eigenvalues is Hermitian:

Verify with HermitianMatrixQ:

A normal matrix with pure imaginary eigenvalues is antihermitian:

Verify with AntihermitianMatrixQ:

A normal matrix whose eigenvalues all have unit modulus is unitary:

Verify with UnitaryMatrixQ:

For any normal matrix, where are eigenvalues of :

Absolute values of eigenvalues of a normal matrix are equal to singular values of the matrix: