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Special Unitary Matrix

A square matrix U is a special unitary matrix if

UU^*=I,
(1)

where I is the identity matrix and U^* is the conjugate transpose matrix, and the determinant is

detU=1.
(2)

The first condition means that U is a unitary matrix, and the second condition provides a restriction beyond a general unitary matrix, which may have determinant e^(itheta) for theta any real number. For example,

1/(sqrt(2))[i i; i -i]
(3)

is a special unitary matrix. A matrix m can be tested to see if it is a special unitary matrix using the Wolfram Language function 

  SpecialUnitaryQ[m_List?MatrixQ] :=
    (Conjugate @ Transpose @ m . m ==
      IdentityMatrix @ Length @ m&& Det[m] == 1)

The special unitary matrices are closed under multiplication and the inverse operation, and therefore form a matrix group called the special unitary group SU(n).

See also

Antihermitian Matrix, Hermitian Inner Product, Special Linear Matrix, Special Unitary Group, Spin Group, Unitary Group Unitary Matrix

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Special Unitary Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SpecialUnitaryMatrix.html

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