set of matrices A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} is said to commute if they commute pairwise, meaning that every pair of matrices in the...

{\displaystyle 2\times 3} . Matrices are commonly related to linear algebra. Notable exceptions include incidence matrices and adjacency matrices in graph theory...

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

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

0, that is, it will "almost never" be singular. Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases...

subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices over L. This is so because...

article. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1;...

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...

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...

alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations...

applied repeatedly and leads to the following sequence of matrices, also called Walsh matrices. H 1 = [ 1 ] , H 2 = [ 1 1 1 − 1 ] , H 4 = [ 1 1 1 1 1 −...

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

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

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...

diagonalizable matrices hold only over an algebraically closed field (such as the complex numbers). In this case, diagonalizable matrices are dense in the...

orthogonal matrix equivalent. Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n × n orthogonal matrices form a group...

all skew-symmetric matrices of a fixed size forms a vector space. The space of n × n {\textstyle n\times n} skew-symmetric matrices has dimension 1 2 n...

restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing...

triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first...

of matrices, often denoted M a t {\displaystyle \mathbf {Mat} } , is the category whose objects are natural numbers and whose morphisms are matrices, with...

p~.} The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone. Some authors...

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...

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...

of Pauli matrices. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of...

expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ. Another factorization is U = [ cos ⁡ ρ − sin...

square matrices, then A ⊗ B and B ⊗ A are even permutation similar, meaning that we can take P = QT. The matrices P and Q are perfect shuffle matrices, called...

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...

two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row...

order n {\displaystyle n} . Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear...

large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms. An important special type of sparse matrices is band matrix...