Unitary matrix

In mathematics, a unitary matrix is a (square) $n\times n$ complex matrix U satisfying the condition

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

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

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

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 $n\times n$ matrix then the following are all equivalent conditions:

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

Properties

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

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

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:

• | det(U) | = 1.
• $U \,$ is invertible, with $U^{-1}=U^{\dagger}$.
• $U^{\dagger}\,$ is also unitary.
• $U \,$ preserves length ("isometry"): $\|Ux\|_2=\|x\|_2$.
• if $U \,$ has complex eigenvalues, they are of modulus 1.
• Eigenspaces are Orthogonal: If matrix is normal then its eigenvectors corresponding to different eigenvalues are orthogonal.
• For any n, the set of all n by n unitary matrices with matrix multiplication forms a group, called U(n).
• Any unit-norm matrix is the average of two unitary matrices. As a consequence, every $n \times n$ matrix is a linear combination of two unitary matrices.

See also

• Orthogonal matrix
• Hermitian matrix
• Symplectic matrix
• Unitary group
• Special unitary group
• Unitary operator
• Matrix decomposition
• Identity matrix

External links

• Weisstein, Eric W., "Unitary Matrix" from MathWorld.
• Ivanova, O. A. (2001), "Unitary matrix", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=U/u095540