pair of matrices in the set commutes. Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed...

commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices....

radius of a product of commuting matrices: if A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} are matrices that all commute, then ρ ( A 1 ⋯ A n ) ≤...

ring, algebraic structures with the commutative property Commuting matrices, sets of matrices whose products do not depend on the order of multiplication...

A} and B {\displaystyle B} : consists only of matrices similar to a diagonal matrix, or has no matrices in it similar to a diagonal matrix, or has exactly...

eigenvalues to form singular values. Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues being unit...

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always...

article. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant...

{\displaystyle 2\times 3} ⁠. Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying...

multiplication of square matrices of a given dimension is a noncommutative operation, except for ⁠ 1 × 1 {\displaystyle 1\times 1} ⁠ matrices. For example: [ 0...

zero if and only if a and b commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then...

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary...

simultaneously diagonalizable because they do not commute. A set consists of commuting normal matrices if and only if it is simultaneously diagonalizable...

\dots ,T_{k}]} in a set of commuting matrices, a weight vector of this algebra is a simultaneous eigenvector of the matrices, while a weight of this algebra...

conventions: matrices are represented by capital letters in bold, e.g. A; vectors in lowercase bold, e.g. a; and entries of vectors and matrices are italic...

\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they...

non-commuting matrices, is justified only by a rejection of unobservable quantities. It introduces the non-commutative multiplication of matrices by physical...

function satisfies ex+y = ex ey. The same is true for commuting matrices. If matrices X and Y commute (meaning that XY = YX), then, e X + Y = e X e Y . {\displaystyle...

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...

definition for 2 × 2 {\displaystyle 2\times 2} -matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant...

scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the...

article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular...

triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices...

between two matrices A {\displaystyle A} and B {\displaystyle B} such that all submatrix products that will be used are defined. Two matrices A {\displaystyle...

n {\displaystyle n\times n} matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously...

C., Ding, J. J., Hsue, W. L., & Chang, K. W. (2008). Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations...

represented by non-commuting matrices, and the matrix elements of bosonic and fermionic particles are ordinary complex numbers and non-commuting Grassmann numbers...

symmetric}}\iff A=A^{\textsf {T}}.} Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix...

multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, similar matrices have the same trace. As a consequence, one can...

{\displaystyle \lambda _{k}} are real. Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral...