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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 .
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.
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model 's electron orbits.
Matrix decomposition. In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Unitary transformation (quantum mechanics) In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle ...
A gate that acts on qubits 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 ).
Unitary group. In mathematics, 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 is a subgroup of the general linear group GL (n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form =, where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition ...