# UnitaryMatrixQ

gives True if m is a unitary matrix, and False otherwise.

# Details and Options • A p×q matrix m is unitary if pq and ConjugateTranspose[m].m is the q×q identity matrix, or pq and m.ConjugateTranspose[m] is the p×p identity matrix.
• UnitaryMatrixQ works for symbolic as well as numerical matrices.
• The following options can be given:
•  Normalized True test if matrix rows are normalized SameTest Automatic function to test equality of expressions Tolerance Automatic tolerance for approximate numbers
• For exact and symbolic matrices, the option SameTest->f indicates that two entries aij and bij are taken to be equal if f[aij,bij] gives True.
• For approximate matrices, the option Tolerance->t can be used to indicate that the norm γ=m.m-In satisfying γt is taken to be zero where In is the identity matrix.

# Examples

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## Basic Examples(1)

Test if a matrix is unitary:

## Scope(4)

A real matrix:

A complex matrix:

A dense matrix:

A sparse matrix:

An approximate MachinePrecision matrix:

An approximate arbitrary-precision matrix:

A matrix with symbolic entries:

The matrix becomes unitary when and :

## Generalizations & Extensions(1)

A matrix can be rectangular:

In this case, matrix rows can be checked to see if they are orthonormal:

A matrix can have more rows than columns:

The columns of the matrix m are orthonormal:

## Options(3)

### Normalized(1)

Symbolic unitary matrix rows are often not normalized to 1:

We can avoid testing if the rows or columns are normalized:

### SameTest(1)

This matrix is unitary for a positive real , but UnitaryMatrixQ gives False:

Use the option SameTest to get the correct answer:

### Tolerance(1)

Generate an orthogonal real-valued matrix with some random perturbation of order 10-14:

q.q is not exactly zero outside the main diagonal:

Adjust the option Tolerance for accepting the matrix as unitary:

Tolerance is applied to the following value:

## Applications(5)

Unitary matrices play an important role in some matrix decompositions:

The inverse of a unitary matrix is unitary:

The inverse of a unitary matrix can be replaced by its conjugate transpose:

Orthogonalize applied to complex vectors generates a unitary matrix:

The matrix does not need to be square:

The matrix is always unitary for any nonzero vector :

It is called a Householder reflection and is used to set to zero selected components of a given column vector :

Check that a matrix drawn from CircularUnitaryMatrixDistribution is unitary:

Check that matrices drawn from CircularOrthogonalMatrixDistribution and CircularSymplecticMatrixDistribution are also unitary:

## Properties & Relations(10)

A matrix is unitary if m.ConjugateTranspose[m]IdentityMatrix[n]:

For an approximate matrix, the identity is approximately true:

Any real-valued orthogonal matrix is unitary:

But a complex unitary matrix is typically not orthogonal:

Dot products of unitary matrices are unitary:

For some matrix functions, a unitary matrix argument gives a unitary matrix:

A unitary matrix is normal:

A unitary matrix has a full set of linear independent eigenvectors:

All eigenvalues of a unitary matrix have the absolute value equal to 1:

The singular values are all 1 for a unitary matrix:

The absolute value of the determinant of a unitary matrix is 1:

The 2-norm of a unitary matrix is always 1:

The matrix exponential MatrixExp of an antihermitian matrix is always unitary: