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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 ...
Gell-Mann matrices. The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU (3) group in the defining representation.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
Then a Givens rotation G can be chosen using the i and j>i rows and columns so as to make the ji element of the product matrix G D A be zero. Since a product of unitary matrices is unitary, the product matrix G D is unitary and so is any product of such matrix pair products.
Schur decomposition. In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.
The group SU (3) is a subgroup of group U (3), the group of all 3×3 unitary matrices. The unitarity condition imposes nine constraint relations on the total 18 degrees of freedom of a 3×3 complex matrix. Thus, the dimension of the U (3) group is 9. Furthermore, multiplying a U by a phase, eiφ leaves the norm invariant.
The Householder matrix has the following properties: it is Hermitian: =,; it is unitary: =,; hence it is involutory: =.; A Householder matrix has eigenvalues .To see this, notice that if is orthogonal to the vector which was used to create the reflector, then =, i.e., is an eigenvalue of multiplicity , since there are independent vectors orthogonal to .
Invariants of tensors. In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial [1] where is the identity operator and are the roots of the polynomial and the eigenvalues of . More broadly,any scalar-valued function is ...