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
$$U^\backslash dagger\; U\; =\; UU^\backslash dagger\; =\; I.$$
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 theirinner 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.
* $\backslash left,\; \backslash det(U)\backslash \; =\; 1$.
* 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 $\backslash Complex^n$ with respect to the usual inner product. In other words, $U^*U\; =\; I$. # The rows of $U$ form an orthonormal basis of $\backslash Complex^n$ with respect to the usual inner product. In other words, $UU^*\; =\; I$. # $U$ is an isometry with respect to the usual norm. That is, $\backslash ,\; Ux\backslash ,\; \_2\; =\; \backslash ,\; x\backslash ,\; \_2$ for all $x\; \backslash in\; \backslash Complex^n$, where $\backslash ,\; x\backslash ,\; \_2\; =\; \backslash sqrt$. # $U$ 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 $$U\; =\; \backslash begin\; a\; \&\; b\; \backslash \backslash \; -e^\; b^*\; \&\; e^\; a^*\; \backslash \backslash \; \backslash end,\; \backslash qquad\; \backslash left,\; a\; \backslash ^2\; +\; \backslash left,\; b\; \backslash ^2\; =\; 1,$$ 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 $$\backslash det(U)\; =\; e^.$$ The sub-group of those elements $U$ with $\backslash det(U)\; =\; 1$ is called the Special unitary group#n = 2, special unitary group SU(2). The matrix can also be written in this alternative form: $$U\; =\; e^\; \backslash begin\; e^\; \backslash cos\; \backslash theta\; \&\; e^\; \backslash sin\; \backslash theta\; \backslash \backslash \; -e^\; \backslash sin\; \backslash theta\; \&\; e^\; \backslash cos\; \backslash theta\; \backslash \backslash \; \backslash end,$$ which, by introducing and , takes the following factorization: $$U\; =\; e^\backslash begin\; e^\; \&\; 0\; \backslash \backslash \; 0\; \&\; e^\; \backslash end\; \backslash begin\; \backslash cos\; \backslash theta\; \&\; \backslash sin\; \backslash theta\; \backslash \backslash \; -\backslash sin\; \backslash theta\; \&\; \backslash cos\; \backslash theta\; \backslash \backslash \; \backslash end\; \backslash begin\; e^\; \&\; 0\; \backslash \backslash \; 0\; \&\; e^\; \backslash end.$$ This expression highlights the relation between unitary matrices and Orthogonal matrix, orthogonal matrices of angle . Another factorization is $$U\; =\; \backslash begin\; \backslash cos\; \backslash alpha\; \&\; -\backslash sin\; \backslash alpha\; \backslash \backslash \; \backslash sin\; \backslash alpha\; \&\; \backslash cos\; \backslash alpha\; \backslash \backslash \; \backslash end\; \backslash begin\; e^\; \&\; 0\; \backslash \backslash \; 0\; \&\; e^\; \backslash end\; \backslash begin\; \backslash cos\; \backslash beta\; \&\; \backslash sin\; \backslash beta\; \backslash \backslash \; -\backslash sin\; \backslash beta\; \&\; \backslash cos\; \backslash beta\; \backslash \backslash \; \backslash end.$$ Many other factorizations of a unitary matrix in basic matrices are possible., page 8See also

* Hermitian matrix * Matrix decomposition * Orthogonal group, Orthogonal group O(''n'') * Orthogonal group, Special orthogonal group SO(''n'') * Orthogonal matrix * Quantum logic gate * Special unitary group, Special Unitary group SU(''n'') * Symplectic matrix * Unitary group , Unitary group U(''n'') * Unitary operatorReferences

External links

* * * {{DEFAULTSORT:Unitary Matrix Matrices Unitary operators