Complex matrix whose conjugate transpose equals its inverse
For matrices with orthogonality over the real number field, see orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see unitarity.
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if
where I 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
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
For real numbers, 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.
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U...
theorem applies: Proposition — A matrix A is normal if and only if there exists a diagonal matrix Λ and a unitarymatrix U such that A = UΛU*. The diagonal...
the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group...
symmetric matrix can be 'diagonalized' using a unitarymatrix: thus if A {\displaystyle A} is a complex symmetric matrix, there is a unitarymatrix U {\displaystyle...
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element...
to: m-by-n matrix A with linearly independent columns Decomposition: A = Q R {\displaystyle A=QR} where Q {\displaystyle Q} is a unitarymatrix of size m-by-m...
m\times m} complex unitarymatrix, Σ {\displaystyle \mathbf {\Sigma } } is an m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative...
only if it is simultaneously diagonalizable by a unitarymatrix; that is, there exists a unitarymatrix U {\displaystyle U} such that U ∗ A U {\displaystyle...
where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint...
the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitarymatrix. Orthogonal...
real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitarymatrix is the identity matrix. The rank of the...
such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Given a field...
Householder matrix is built through the identity matrix) Square root of a 2 by 2 identity matrixUnitarymatrix Zero matrix "Identity matrix: intro to identity...
matrix representation. According to the documentation, in LAPACK the unitarymatrix representation is optimized. Some languages such as Java store matrices...
complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitarymatrix and P {\displaystyle...
by unitary matrices. A gate that acts on n {\displaystyle n} qubits is represented by a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} unitarymatrix, and...
P=P^{*}} , it is unitary: P − 1 = P ∗ {\textstyle P^{-1}=P^{*}} , hence it is involutory: P = P − 1 {\textstyle P=P^{-1}} . A Householder matrix has eigenvalues...
a matrix decomposition of a matrix A {\displaystyle A} into a unitarymatrix P {\displaystyle P} and a Hessenberg matrix H {\displaystyle H} such that...
^{-1}.} A square complex matrix whose transpose is equal to its conjugate inverse is called a unitarymatrix; that is, A is unitary if A T = A − 1 ¯ . {\displaystyle...
determinant +1. The complex analogue of an orthogonal matrix is a unitarymatrix. A real or complex square matrix A {\displaystyle A} is called normal if A ∗ A...
chaotic behaviour are described by random matrix theory. In quantum optics, transformations described by random unitary matrices are crucial for demonstrating...
similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitarymatrix then A and...
real symmetric matrix and more generally a Hermitian matrix, and a unitarymatrix, is never defective; more generally, a normal matrix (which includes...
{\displaystyle k\times n} matrix B {\displaystyle B} and if Q {\displaystyle Q} is any unitary k × k {\displaystyle k\times k} matrix (meaning Q ∗ Q = Q Q...
Global Information
