Unitary Matrix

A square matrix U is a unitary matrix if


where U^(H) denotes the conjugate transpose and U^(-1) is the matrix inverse. For example,

 A=[2^(-1/2) 2^(-1/2) 0; -2^(-1/2)i 2^(-1/2)i 0; 0 0 i]

is a unitary matrix.

Unitary matrices leave the length of a complex vector unchanged.

For real matrices, unitary is the same as orthogonal. In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of a unitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.

A matrix m can be tested to see if it is unitary in the Wolfram Language using UnitaryMatrixQ[m].

The definition of a unitary matrix guarantees that


where I is the identity matrix. In particular, a unitary matrix is always invertible, and U^(-1)=U^(H). Note that transpose is a much simpler computation than inverse. A similarity transformation of a Hermitian matrix with a unitary matrix gives


Unitary matrices are normal matrices. If M is a unitary matrix, then the permanent


(Minc 1978, p. 25, Vardi 1991).

The unitary matrices are precisely those matrices which preserve the Hermitian inner product


Also, the norm of the determinant of U is |detU|=1. Unlike the orthogonal matrices, the unitary matrices are connected. If detU=1 then U is a special unitary matrix.

The product of two unitary matrices is another unitary matrix. The inverse of a unitary matrix is another unitary matrix, and identity matrices are unitary. Hence the set of unitary matrices form a group, called the unitary group.

See also

Antihermitian Matrix, Clifford Algebra, Conjugate Transpose, Group Representation, Hermitian Inner Product, Hermitian Matrix, Normal Matrix, Orthogonal Group, Permanent, Special Unitary Matrix, Spin Group, Symmetric Matrix, Unimodular Matrix, Unit Matrix, Unitary Group

Portions of this entry contributed by Todd Rowland

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Minc, H. §3.1 in Permanents. Reading, MA: Addison-Wesley, 1978.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, 1991.

Referenced on Wolfram|Alpha

Unitary Matrix

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Unitary Matrix." From MathWorld--A Wolfram Web Resource.

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