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

{\displaystyle 2\times 3} ⁠. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations (for example...

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

product: ch. 5  or Schur product) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding...

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

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

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

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...

Equivalently, the set of singular matrices is closed and nowhere dense in the space of n-by-n matrices. In practice, however, non-invertible matrices...

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

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...

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

definiteness, permitting the matrices to be non-symmetric or non-Hermitian. The properties of these generalized definite matrices are explored in § Extension...

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...

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...

matrices arise in the study of operator algebras and the theory of quantum computation. Butson-type Hadamard matrices are complex Hadamard matrices in...

triangle matrices combined contain n ( n + 1 ) {\displaystyle n(n+1)} coefficients, therefore n {\displaystyle n} coefficients of matrices LU are not...

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

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...

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...

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

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...

the space of skew-symmetric 3 x 3 matrices over R {\displaystyle \mathbb {R} } . A basis is given by the three matrices F 1 = ( 0 0 0 0 0 − 1 0 1 0 ) ,...

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

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...

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...

of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging...

product of two unimodular matrices Other examples include: Pascal matrices Permutation matrices the three transformation matrices in the ternary tree of...