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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 .
A set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I 2 identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices. Redheffer matrix: Encodes a Dirichlet convolution. Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function.
A gate that acts on qubits (a register) is represented by a unitary matrix, and the set of all such gates with the group operation of matrix multiplication [a] is the unitary group U(2 n). [2] The quantum states that the gates act upon are unit vectors in 2 n {\displaystyle 2^{n}} complex dimensions, with the complex Euclidean norm (the 2-norm ).
Toggle Examples subsection. 2.1 Two-point. ... a DFT matrix is an expression of a discrete Fourier transform ... choice here makes the resulting DFT matrix unitary, ...
More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. [4] They are the unitary operators on R n.
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication.
The general unitary group, also called the group of unitary similitudes, consists of all matrices A such that A ∗ A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Unitary groups may also be defined over fields other than the complex ...
Singular value decomposition expresses any matrix A as a product UDV ∗, where U and V are unitary matrices and D is a diagonal matrix. An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.