unitary matrix

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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (math ...
, a
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is unitary if its
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
is also its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
, that is, if $U^* U = UU^* = UU^ = I,$ where is the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I'n'', or simply by ''I'' if the size is immaterial or can be trivially determined by ...

. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the
Hermitian adjointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of a matrix and is denoted by a
dagger A dagger is a knife A knife (plural knives; from Old Norse'' knifr'', "knife, dirk") is a tool or weapon with a cutting edge or blade, often attached to a handle or hilt. One of the earliest tools used by humanity, knives appeared at least ...
(†), so the equation above becomes $U^\dagger U = UU^\dagger = I.$ The real analogue of a unitary matrix is an
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is :Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose In linear algebra, t ...
. Unitary matrices have significant importance in quantum mechanics because they preserve
norms Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanato ...
, and thus,
probability amplitude In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
s.

# Properties

For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
; that is, . * is normal ($U^* U = UU^*$). * is
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrixIn linear algebra, an ''n''-by-''n'' square matrix is called invertible (also ...
; that is, is unitarily similar 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. * $\left, \det\left(U\right)\ = 1$. * Its Eigenvector#Eigenspaces of a matrix, eigenspaces are orthogonal. * can be written as , where indicates the matrix exponential, is the imaginary unit, and is a Hermitian matrix. For any nonnegative integer ''n'', the set of all ''n'' × ''n'' unitary matrices with matrix multiplication forms a group (mathematics), group, called the unitary group U(''n''). Any square matrix with unit Euclidean norm is the average of two unitary matrices.

# Equivalent conditions

If ''U'' is a square, complex matrix, then the following conditions are equivalent: # $U$ is unitary. # $U^*$ is unitary. # $U$ is invertible with $U^ = U^*$. # The columns of $U$ form an orthonormal basis of $\Complex^n$ with respect to the usual inner product. In other words, $U^*U = I$. # The rows of $U$ form an orthonormal basis of $\Complex^n$ with respect to the usual inner product. In other words, $UU^* = I$. # $U$ is an isometry with respect to the usual norm. That is, $\, Ux\, _2 = \, x\, _2$ for all $x \in \Complex^n$, where $\, x\, _2 = \sqrt$. # $U$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $U$) with eigenvalues lying on the unit circle.

# Elementary constructions

## 2 × 2 unitary matrix

The general expression of a unitary matrix is $U = \begin a & b \\ -e^ b^* & e^ a^* \\ \end, \qquad \left, a \^2 + \left, b \^2 = 1,$ which depends on 4 real parameters (the phase of , the phase of , the relative magnitude between and , and the angle ). The determinant of such a matrix is $\det(U) = e^.$ The sub-group of those elements $U$ with $\det\left(U\right) = 1$ is called the Special unitary group#n = 2, special unitary group SU(2). The matrix can also be written in this alternative form: $U = e^ \begin e^ \cos \theta & e^ \sin \theta \\ -e^ \sin \theta & e^ \cos \theta \\ \end,$ which, by introducing and , takes the following factorization: $U = e^\begin e^ & 0 \\ 0 & e^ \end \begin \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end \begin e^ & 0 \\ 0 & e^ \end.$ This expression highlights the relation between unitary matrices and Orthogonal matrix, orthogonal matrices of angle . Another factorization is $U = \begin \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end \begin e^ & 0 \\ 0 & e^ \end \begin \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \\ \end.$ Many other factorizations of a unitary matrix in basic matrices are possible., page 8