Unitary matrix

 ( Matrices ) 

In linear algebra, an invertible Complex number square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if

$$U^*\; U\; =\; UU^*\; =\; I,$$

where is the identity matrix.

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (), so the equation above is written

$$U^\backslash dagger\; U\; =\; UU^\backslash dagger\; =\; I.$$

A complex matrix is special unitary if it is unitary and its matrix determinant equals .

For , the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

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Properties For any unitary matrix of finite size, the following hold:

• Given two complex vectors and , multiplication by preserves their inner product; that is, .
• is normal matrix ($U^*\; U\; =\; UU^*$).
• is diagonalizable; that is, is similar matrix to a diagonal matrix, as a consequence of the spectral theorem. Thus, has a decomposition of the form $U\; =\; VDV^*,$ where is unitary, and is diagonal and unitary.
• The eigenvalues of $U$ lie on the unit circle, as does $\backslash det(U)$.
• The of $U$ are orthogonal.
• can be written as , where indicates the matrix exponential, is the imaginary unit, and is a Hermitian matrix.

For any nonnegative integer , the set of all unitary matrices with matrix multiplication forms a group, called the unitary group .

Every square matrix with unit Euclidean norm is the average of two unitary matrices.

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Equivalent conditions If U is a square, complex matrix, then the following conditions are equivalent:

(2025). 9781139020411, Cambridge University Press. ISBN 9781139020411

1. $U$ is unitary.
2. $U^*$ is unitary.
3. $U$ is invertible with $U^\{-1\}\; =\; U^*$.
4. The columns of $U$ form an orthonormal basis of $\backslash Complex^n$ with respect to the usual inner product. In other words, $U^*U\; =\; I$.
5. The rows of $U$ form an orthonormal basis of $\backslash Complex^n$ with respect to the usual inner product. In other words, $UU^*\; =\; I$.
6. $U$ is an isometry with respect to the usual norm. That is, $\backslash |Ux\backslash |\_2\; =\; \backslash |x\backslash |\_2$ for all $x\; \backslash in\; \backslash Complex^n$, where $\backslash |x\backslash |\_2\; =\; \backslash sqrt\{\backslash sum\_\{i=1\}^n\; |x\_i|^2\}$.
7. $U$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $U$) with eigenvalues lying on the unit circle.

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Elementary constructions

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2 × 2 unitary matrix One general expression of a unitary matrix is

$$U\; =\; \backslash begin\{bmatrix\}$$

a & b \\
-e^{i\varphi} b^* & e^{i\varphi} a^* \\
     

\end{bmatrix}, \qquad \left| a \right|^2 + \left| b \right|^2 = 1\ ,

which depends on 4 real parameters (the phase of , the phase of , the relative magnitude between and , and the angle ). The form is configured so the determinant of such a matrix is $$\backslash det(U)\; =\; e^\{i\; \backslash varphi\}\; ~.$$

The sub-group of those elements $\backslash \; U\backslash$ with $\backslash \; \backslash det(U)\; =\; 1\backslash$ is called the special unitary group SU(2).

Among several alternative forms, the matrix can be written in this form: $$\backslash \; U\; =\; e^\{i\backslash varphi\; /\; 2\}\; \backslash begin\{bmatrix\}$$

e^{i\alpha} \cos \theta & e^{i\beta} \sin \theta \\
-e^{-i\beta} \sin \theta & e^{-i\alpha} \cos \theta \\
     

\end{bmatrix}\ ,

where $\backslash \; e^\{i\backslash alpha\}\; \backslash cos\; \backslash theta\; =\; a\backslash$ and $\backslash \; e^\{i\backslash beta\}\; \backslash sin\; \backslash theta\; =\; b\backslash \; ,$ above, and the angles $\backslash \; \backslash varphi,\; \backslash alpha,\; \backslash beta,\; \backslash theta\backslash$ can take any values.

By introducing $\backslash \; \backslash alpha\; =\; \backslash psi\; +\; \backslash delta\backslash$ and $\backslash \; \backslash beta\; =\; \backslash psi\; -\; \backslash delta\backslash \; ,$ has the following factorization:

$$U\; =\; e^\{i\backslash varphi\; /2\}\; \backslash begin\{bmatrix\}$$

e^{i\psi} & 0 \\
0 & e^{-i\psi}
     

\end{bmatrix} \begin{bmatrix}

\cos \theta  & \sin \theta \\
-\sin \theta & \cos \theta \\
     

\end{bmatrix} \begin{bmatrix}

e^{i\delta} & 0 \\
0 & e^{-i\delta}
     

\end{bmatrix} ~.

This expression highlights the relation between unitary matrices and orthogonal matrices of angle .

Another factorization is

$$U\; =\; \backslash begin\{bmatrix\}$$

\cos \rho  &   -\sin \rho \\
\sin \rho  &   \;\cos \rho \\
     

\end{bmatrix} \begin{bmatrix}

e^{i\xi} & 0 \\
0 & e^{i\zeta}
     

\end{bmatrix} \begin{bmatrix}

\;\cos \sigma  &   \sin \sigma \\
-\sin \sigma   &   \cos \sigma \\
     

\end{bmatrix} ~.

Many other factorizations of a unitary matrix in basic matrices are possible.

(2025). 9781107002173, Cambridge University Press. . ISBN 9781107002173

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See also

• Hermitian matrix
• Skew-Hermitian matrix
• Matrix decomposition
• Orthogonal group
• Orthogonal group
• Orthogonal matrix
• Semi-orthogonal matrix
• Quantum logic gate
• Special Unitary group SU( n)
• Symplectic matrix
• Unitary group
• Unitary operator

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External links

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