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...
9 KB (1,390 words) - 04:09, 29 October 2023
Matrix (mathematics) (redirect from Real matrices)
Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article...
105 KB (13,048 words) - 22:10, 26 May 2024
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually...
45 KB (7,429 words) - 17:47, 26 May 2024
Invertible matrix (redirect from Invertible matrices)
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...
46 KB (6,931 words) - 08:34, 29 May 2024
\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they...
60 KB (7,209 words) - 14:01, 14 March 2024
Rotation matrix (redirect from Rotation matrices)
article. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1;...
99 KB (15,019 words) - 04:42, 13 May 2024
Hermitian matrix (redirect from Hermitian matrices)
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...
19 KB (2,992 words) - 02:35, 16 May 2024
Matrix similarity (redirect from Similar matrices)
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...
8 KB (1,081 words) - 22:27, 10 April 2024
Matrix multiplication (redirect from Multiplying matrices)
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...
40 KB (6,456 words) - 13:03, 8 May 2024
Transformation matrix (redirect from Transformation Matrices)
alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations...
24 KB (3,826 words) - 19:39, 10 March 2024
Diagonalizable matrix (redirect from Diagonalizable matrices)
diagonalizable matrices hold only over an algebraically closed field (such as the complex numbers). In this case, diagonalizable matrices are dense in the...
23 KB (4,044 words) - 13:36, 12 April 2024
article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular...
31 KB (1,336 words) - 00:12, 30 November 2023
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...
10 KB (1,387 words) - 12:18, 25 February 2024
Normal matrix (redirect from Normal matrices)
eigenvalues to form singular values. Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues being unit...
13 KB (1,630 words) - 22:34, 6 December 2023
Block matrix (redirect from Block 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...
29 KB (4,772 words) - 21:59, 25 April 2024
Hadamard matrix (redirect from Hadamard matrices)
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 −...
25 KB (3,078 words) - 00:15, 27 May 2024
Definite matrix (redirect from Positive-definite matrices)
at 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...
47 KB (8,431 words) - 10:11, 10 April 2024
Triangular matrix (redirect from Triangular matrices)
triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first...
21 KB (3,152 words) - 16:41, 28 April 2024
Orthogonal matrix (redirect from Orthogonal matrices)
orthogonal matrix equivalent. Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n × n orthogonal matrices form a group...
36 KB (4,802 words) - 16:45, 22 May 2024
Skew-symmetric matrix (redirect from Skew-symmetric matrices)
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...
18 KB (3,543 words) - 22:03, 30 October 2023
Diagonal matrix (redirect from Diagonal matrices)
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...
17 KB (2,466 words) - 02:48, 13 May 2024
Matrix equivalence (redirect from Equivalent matrices)
restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing...
4 KB (619 words) - 23:34, 8 February 2024
Unitary matrix (redirect from Unitary matrices)
expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ. Another factorization is U = [ cos ρ − sin...
10 KB (1,307 words) - 23:15, 10 May 2024
Square matrix (redirect from Square matrices)
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...
16 KB (1,831 words) - 01:25, 27 April 2024
Kronecker product (redirect from Tensor product of matrices)
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. The...
40 KB (6,118 words) - 19:59, 25 April 2024
Row equivalence (redirect from Equivalent Matrix)
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...
6 KB (817 words) - 08:48, 9 February 2023
Determinant (section Two by two matrices)
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...
89 KB (14,131 words) - 13:35, 28 May 2024
Sparse matrix (redirect from Sparse matrices)
large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms. An important special type of sparse matrices is band matrix...
25 KB (3,182 words) - 05:02, 7 April 2024
Circulant matrix (redirect from Circulant matrices)
{\displaystyle j} even or odd. Symmetric circulant matrices belong to the class of bisymmetric matrices. The complex version of the circulant matrix, ubiquitous...
15 KB (2,755 words) - 13:33, 25 February 2024
Matrix ring (redirect from Ring of matrices)
a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries...
13 KB (1,812 words) - 23:17, 26 March 2024