A square matrix is a unitary matrix if
(1)
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where denotes the conjugate transpose and is the matrix inverse. For example,
(2)
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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 can be tested to see if it is unitary in the Wolfram Language using UnitaryMatrixQ[m].
The definition of a unitary matrix guarantees that
(3)
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where is the identity matrix. In particular, a unitary matrix is always invertible, and . Note that transpose is a much simpler computation than inverse. A similarity transformation of a Hermitian matrix with a unitary matrix gives
(4)
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(5)
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(6)
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(7)
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(8)
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Unitary matrices are normal matrices. If is a unitary matrix, then the permanent
(9)
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(Minc 1978, p. 25, Vardi 1991).
The unitary matrices are precisely those matrices which preserve the Hermitian inner product
(10)
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Also, the norm of the determinant of is . Unlike the orthogonal matrices, the unitary matrices are connected. If then 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.