Unitary matrix: Definition from Answers.com

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

$U^{*} U = UU^{*} = I_n\,$

where $I_n\,$ is the identity matrix in n dimensions and $U^{*} \,$ is the conjugate transpose (also called the Hermitian adjoint) of $U$ . Note this condition says that a matrix $U$ is unitary if and only if it has an inverse which is equal to its conjugate transpose $U^{*} \,$

$U^{-1} = U^{*} \,\;$

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

$\langle Gx, Gy \rangle = \langle x, y \rangle$

so also a unitary matrix $U$ satisfies

$\langle Ux, Uy \rangle = \langle x, y \rangle$

for all complex vectors x and y, where $\langle\cdot,\cdot\rangle$ stands now for the standard inner product on $\mathbb{C}^n$ .

If $U \,$ is an n by n matrix then the following are all equivalent conditions:

$U \,$ is unitary
$U^{*} \,$ is unitary
the columns of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
the rows of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
$U \,$ is an isometry with respect to the norm from this inner product
U is a normal matrix with eigenvalues lying on the unit circle.

1 Properties of unitary matrices
2 See also
3 References
4 External links

Properties of unitary matrices

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

$U = V\Sigma V^{*}\;$

where

V

is unitary, and

Σ

is diagonal and unitary. That is, a unitary matrix is diagonalizable by a unitary matrix.

For any unitary matrix $U$ , the following hold:

$U$ is invertible.
$U - 1 = U * .$
$| det(U) | = 1.$
$U *$ is unitary.
$U$ preserves length $\|Ux\|_2=\|x\|_2.$
$U$ has complex eigenvalues of modulus 1. ^[1]
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).
For any n, the set of all n by n unitary matrices with matrix multiplication forms a group, called U(n).
Any matrix is the average of two unitary matrices. As a consequence, every $n \times n$ matrix $M$ is a linear combination of two unitary matrices.^[2]

References

^ R. Shankar, Principles of Quantum Mechanics, 2nd Ed., pg. 39.
^ Chi-Kwong Li and Edward Poon, Additive Decomposition of Real Matrices, pg. 1.

External links

Rowland, Todd, "Unitary Matrix" from MathWorld.
Ivanova, O.A. (2001), "Unitary matrix", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/U/u095540.htm

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Unitary matrix

Contents

Properties of unitary matrices

See also

References

External links