Unitary matrix

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In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition

<math>U^* U = UU^* = I_n\,</math>

where <math>I_n\,</math> is the identity matrix and <math>U^* \,</math> is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose <math>U^* \,</math>.

A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors,

<math>\langle Gx, Gy \rangle = \langle x, y \rangle</math>

so also a unitary matrix U satisfies

<math>\langle Ux, Uy \rangle = \langle x, y \rangle</math>

for all complex vectors x and y, where <.,.> stands now for the standard inner product on Cⁿ. If <math>A \,</math> is an n by n matrix then the following are all equivalent conditions:

1. <math>A \,</math> is unitary
2. <math>A^* \,</math> is unitary
3. the columns of <math>A \,</math> form an orthonormal basis of Cⁿ with respect to this inner product
4. the rows of <math>A \,</math> form an orthonormal basis of Cⁿ with respect to this inner product
5. <math>A \,</math> is an isometry with respect to the norm from this inner product

It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e. they lie on the unit circle centered at 0 in the complex plane). The same is true for the determinant.

All unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix U has a decomposition of the form

<math>U = V\Sigma V^*</math>

where V is unitary, and <math>\Sigma</math> is diagonal and unitary.

For any n, the set of all n by n unitary matrices with matrix multiplication form a group.

A unitary matrix is called special if its determinant is 1.

See also

• orthogonal matrix
• symplectic matrix
• unitary group
• special unitary group
• unitary operatorde:Unitäre Matrix

fr:Matrice unitaire it:Matrice unitaria he:יוניטריות ja:ユニタリ作用素 pl:Macierz unitarna