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 $\Sigma$ 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

• Rowland, Todd. "Unitary Matrix". MathWorld. {{cite web}}: More than one of |author= and |last= specified (help)
• Ivanova, O. A. (2001) [1994], "Unitary matrix", Encyclopedia of Mathematics, EMS Press