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Mathematical concept in algebra
pair of matrices in the set commutes. Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed
Commuting_matrices
Special kind of square matrix
commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.
Triangular_matrix
Array of numbers
matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations (for example
Matrix_(mathematics)
Concept in linear algebra
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
Matrix_pencil
Matrix defined using smaller matrices called blocks
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
Block_matrix
Topics referred to by the same term
ring, algebraic structures with the commutative property Commuting matrices, sets of matrices whose products do not depend on the order of multiplication
Commute
Largest absolute value of an operator's eigenvalues
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 ) ≤
Spectral_radius
Theorem representing a solvable Lie algebra
triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices
Lie's_theorem
Generators of the Clifford algebra for relativistic quantum mechanics
\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they
Gamma_matrices
Matrices similar to diagonal matrices
simultaneously diagonalizable because they do not commute. A set consists of commuting normal matrices if and only if it is simultaneously diagonalizable
Diagonalizable_matrix
Matrices important in quantum mechanics and the study of spin
mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory
Pauli_matrices
Operation measuring the failure of two entities to commute
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
Commutator
Mathematical operation in linear 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
Matrix_multiplication
Matrix that commutes with its conjugate transpose
eigenvalues to form singular values. Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues being unit
Normal_matrix
Matrix operation generalizing exponentiation of scalar numbers
function satisfies ea+b = ea eb. 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
Matrix_exponential
Basis for the SU(3) Lie algebra
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
Gell-Mann_matrices
Real square matrix whose columns and rows are orthogonal unit vectors
orthogonal matrices, under multiplication, forms the group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant
Orthogonal_matrix
Matrix whose only nonzero elements are on its main diagonal
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
Diagonal_matrix
Matrix representing a Euclidean rotation
article. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant
Rotation_matrix
A matrix canonical form
generated by two commuting n × n {\displaystyle n\times n} matrices has dimension at most n {\displaystyle n} . A set of finite matrices is said to be approximately
Weyr_canonical_form
Function in discrete mathematics
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
Discrete_Fourier_transform
German physicist (1901–1976)
non-commuting matrices, is justified only by a rejection of unobservable quantities. It introduces the non-commutative multiplication of matrices by physical
Werner_Heisenberg
Matrix equal to its conjugate-transpose
k {\displaystyle k} . Hermitian matrices can be understood as the complex generalization of symmetric real matrices. Using the notation A T {\displaystyle
Hermitian_matrix
Matrix with shifting rows
Toeplitz matrices, and underlies the effectiveness of DFT-based spectral density estimation for stationary processes. Toeplitz matrices commute asymptotically
Toeplitz_matrix
Property of some mathematical operations
multiplication of square matrices of a given dimension is a noncommutative operation, except for 1 × 1 {\displaystyle 1\times 1} matrices. For example: [ 0
Commutative_property
Matrix equal to its transpose
symmetric}}\iff A=A^{\textsf {T}}.} Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix
Symmetric_matrix
In mathematics, invariant of square 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
Determinant
Most widely known generalized inverse of a matrix
established. Since for invertible matrices the pseudoinverse equals the usual inverse, only examples of non-invertible matrices are considered below. For A
Moore–Penrose_inverse
. {\displaystyle e^{a+b}=e^{a}e^{b}.} If we replace a and b with commuting matrices A and B, then the same inequality e A + B = e A e B {\displaystyle
Golden–Thompson_inequality
Concept in Lie algebra representation theory
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
Weight (representation theory)
Weight_(representation_theory)
Study of abstract algebraic structures
\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
Algebra_representation
article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular
List_of_named_matrices
American physicist
represented by non-commuting matrices, and the matrix elements of bosonic and fermionic particles are ordinary complex numbers and non-commuting Grassmann numbers
Stephen_L._Adler
Square matrices satisfy their characteristic equation
{\displaystyle 2\times 2} complex matrices. Cayley in 1858 stated the result for 3 × 3 {\displaystyle 3\times 3} and smaller matrices, but only published a proof
Cayley–Hamilton_theorem
Manifold with inversion symmetry
form by an orthogonal matrix and this can be done simultaneously for commuting matrices. Type IVn: by a transformation in SO(n) × SO(2) any vector can be
Hermitian_symmetric_space
Form of a matrix
{\displaystyle \lambda _{k}} are real. Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral
Skew-symmetric_matrix
Matrix with exactly one 1 per row and column
P^{-1}=P^{\mathsf {T}}} . Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative. There are two
Permutation_matrix
Algebraic structure used in analysis
of all n × n {\displaystyle n\times n} matrices. The Lie bracket is defined to be the commutator of matrices (or linear maps), [ X , Y ] = X Y − Y X
Lie_algebra
Sum of elements on the main diagonal
multiplicities), see below. 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
Trace_(linear_algebra)
16-element matrix group
products of Pauli matrices, including the identity. The single-qubit Pauli group is a 16-element matrix group, consisting of the 4 Pauli matrices each with 4
Pauli_group
Matrix factorisation in mathematics
consider an eigenspace VA. Then VA is invariant under all matrices in {Ai}. Therefore, all matrices in {Ai} must share one common eigenvector in VA. Induction
Schur_decomposition
Gamma matrices for arbitrary Clifford algebras
mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Subject area in mathematics
K_{1}(A)\cdots \ .} There is a pairing on K1 with values in K2. Given commuting matrices X and Y over A, take elements x and y in the Steinberg group with
Algebraic_K-theory
Numerical linear algebra algorithm
generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices. Since singular values of a real matrix
Jacobi_eigenvalue_algorithm
n-by-n matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at n-by-n matrices. That
Central_polynomial
Theorem in linear algebra
and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices with exclusively non-negative real numbers
Perron–Frobenius_theorem
Mathematical description of fermions
{\displaystyle \sigma _{i}} are the Pauli matrices and α {\displaystyle \alpha } is the vector made of gamma matrices α = γ t ( γ x , γ y , γ z ) {\displaystyle
Dirac_spinor
Algebraic term
{U} _{n}} of upper-triangular matrices with 1 {\displaystyle 1} 's along the diagonal, so they are the group of matrices U n = { [ 1 ∗ ⋯ ∗ ∗ 0 1 ⋯ ∗ ∗
Unipotent
Mathematical ring whose elements are 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
Matrix_ring
matrix commutes with all matrices of the representation then it is a (complex) multiple of the identity matrix, that is, the set of commuting matrices is
Corepresentations of unitary and antiunitary groups
Corepresentations_of_unitary_and_antiunitary_groups
Group of rotations in 3 dimensions
identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3)
3D_rotation_group
n-by-n matrices X 1 , … , X m {\displaystyle X_{1},\dots ,X_{m}} , where ( X l ) i j {\displaystyle (X_{l})_{ij}} are matrix entries and commute by definition
Generic_matrix_ring
Matrix whose entries are all minors of another matrix
adjugate matrices appear when computing determinants of linear combinations of matrices. It is elementary to check that if A and B are n × n matrices then
Compound_matrix
Function that maps matrices to matrices
used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential
Analytic_function_of_a_matrix
Four-dimensional number system
numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion
Quaternion
Group of 𝑛 × 𝑛 invertible matrices
invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again
General_linear_group
Australian and American mathematician (born 1975)
initiated the study of random matrices and their eigenvalues. Wigner studied the case of hermitian and symmetric matrices, proving a "semicircle law" for
Terence_Tao
Integral expressing the amount of overlap of one function as it is shifted over another
evolving of count sketch properties). This can be generalized for appropriate matrices A , B {\displaystyle \mathbf {A} ,\mathbf {B} } : W ( ( A x ) ∗ ( B y )
Convolution
Numerical methods for matrix eigenvalue calculation
matrices. While there is no simple algorithm to directly calculate eigenvalues for general matrices, there are numerous special classes of matrices where
Eigenvalue_algorithm
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following matrix:
Jordan_normal_form
Relativistic quantum mechanical wave equation
matrices as a candidate, but then showed these would not work since it is impossible to find a set of four 2 × 2 {\displaystyle 2\times 2} matrices that
Dirac_equation
Vector space equipped with a bilinear product
(but not excluded, either). Given an integer n, the ring of real square matrices of order n is an example of an associative algebra over the field of real
Algebra_over_a_field
Element of a unital algebra over the field of real numbers
real matrices were found isomorphic to coquaternions. Soon the matrix paradigm began to explain several others as they were represented by matrices and
Hypercomplex_number
Type of matrix representation
definition A = U P {\displaystyle A=UP} may be extended to rectangular matrices A ∈ C m × n {\displaystyle A\in \mathbb {C} ^{m\times n}} by requiring
Polar_decomposition
Formula of matrix exponentials
replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to
Lie_product_formula
Formulation of quantum mechanics
the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture
Matrix_mechanics
Group that is also a differentiable manifold with group operations that are smooth
connected components containing matrices with positive and negative determinants, respectively. The rotation matrices form a subgroup of GL ( 2 ,
Lie_group
Generating function in integrable systems
order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters
Tau function (integrable systems)
Tau_function_(integrable_systems)
Non-tensorial representation of the spin group
needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar
Spinor
Group of unitary matrices
1 {\displaystyle q+1} and consists of the scalar matrices that are unitary, that is those matrices c I V {\displaystyle cI_{V}} with c q + 1 = 1 {\displaystyle
Unitary_group
Mathematical operation
the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex
Cayley_transform
Motion of a certain space that preserves at least one point
symmetry law of nature. The complex-valued matrices analogous to real orthogonal matrices are the unitary matrices U ( n ) {\displaystyle \mathrm {U} (n)}
Rotation_(mathematics)
In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative
Manin_matrix
Group of symmetries of an n-dimensional hypercube
of n × n {\displaystyle n\times n} orthogonal matrices. It consists exactly of those orthogonal matrices whose entries are all integers. For n ≥ 2 {\displaystyle
Hyperoctahedral_group
Mathematical model of magnetism
terms of the newly defined Pauli matrices with tildes, which obey the same algebraic relations as the original Pauli matrices, the Hamiltonian is simply H
Transverse-field_Ising_model
Mathematical operation on invertible matrices
all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads
Logarithm_of_a_matrix
Matrix symmetric about its center
centrosymmetric and skew-centrosymmetric matrices. Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia
Centrosymmetric_matrix
Mathematical weight device
orthogonal designs can be discovered by way of weighing matrices. In this display of weighing matrices the symbol − {\displaystyle -} is used to represent
Weighing_matrix
Intrinsic quantum property of particles
all n-fold tensor products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices R ^ ( θ , n ^ ) = e i θ 2 n ^ ⋅ σ =
Spin_(physics)
Term in quantum mechanics
matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices,
Fidelity_of_quantum_states
Generalization of additive and multiplicative inverses
invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings. In the case of integer matrices (that is, matrices with
Inverse_element
Square matrix which is its own inverse
interchanged rows; S is a signature matrix. Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear
Involutory_matrix
Branch of mathematics that studies abstract algebraic structures
in terms of invertible matrices. Matrix addition and multiplication make the set of all n × n {\displaystyle n\times n} matrices into an associative algebra
Representation_theory
Square matrix symmetric about both its diagonal and anti-diagonal
1137/S0895479801386730. Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia. 32 (2): 631–644. doi:10
Bisymmetric_matrix
Mathematical entity to describe the probability of each possible measurement on a system
on that Hilbert space, and instead are commonly represented as density matrices. Common examples of quantum states are the wave functions describing position
Quantum_state
Basic circuit in quantum computing
omitted. All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices. Positive integer exponents are equivalent
Quantum_logic_gate
Algebraic structure with addition and multiplication
if R is the ring of all square matrices of size n over a field, then R× consists of the set of all invertible matrices of size n, and is called the general
Ring_(mathematics)
Formula in Lie theory
{\displaystyle S=\mathbb {R} [[X,Y]]} of all non-commuting formal power series with real coefficients in the non-commuting variables X and Y. There is a ring homomorphism
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra
Fuzzy_sphere
Part of the theory of modular forms
Q) generated by Γ0(N) together with the matrices We; let Γ0(N)+ denote its quotient by positive scalar matrices. Then Γ0(N) is a normal subgroup of Γ0(N)+
Atkin–Lehner_theory
Identities involving spinor bilinears
of commuting spinors, otherwise, as is the case of fermions in physics, all coefficients change signs. For example, under the assumption of commuting spinors
Fierz_identity
Family of linear transformations
Throughout, italic non-bold capital letters are 4 × 4 matrices, while non-italic bold letters are 3 × 3 matrices. Writing the coordinates in column vectors and
Lorentz_transformation
Special orthogonal group
are the eigenvalues of A. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices A1 and A2 by Rodrigues' rotation formula and the
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
Set of elements that commute with every element of a group
G, is all of G. The center of the Heisenberg group, H, is the set of matrices of the form: ( 1 0 z 0 1 0 0 0 1 ) {\displaystyle {\begin{pmatrix}1&0&
Center_(group_theory)
States that the algebra of n by n matrices satisfies a certain identity of degree 2n
algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It was
Amitsur–Levitzki_theorem
Properties underlying modern physics
J(n). The three J(m) matrices are each (2m + 1)×(2m + 1) square matrices, and the three J(n) are each (2n + 1)×(2n + 1) square matrices. The integers or half-integers
Symmetry_in_quantum_mechanics
eigenvalues of sums or products of independent random matrices. Through its applications to random matrices, free convolution has some strong connections with
Free_convolution
Concept in Hlibert spaces mathematics
involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices. Let
Trace_inequality
equivalent characterization of EP matrices is that the range of A is orthogonal to the nullspace of A. Thus, EP matrices are also known as RPN (Range Perpendicular
EP_matrix
Representation of angular momentum tensor product states important to physics
down, and strange. The special unitary group SU is the group of unitary matrices whose determinant is equal to 1. This set is closed under matrix multiplication
Clebsch–Gordan coefficients for SU(3)
Clebsch–Gordan_coefficients_for_SU(3)
COMMUTING MATRICES
COMMUTING MATRICES
COMMUTING MATRICES
COMMUTING MATRICES
Boy/Male
Indian, Punjabi, Sikh
Strong; Brave
Girl/Female
American, Australian, British, Chinese, Danish, Dutch, English, Finnish, French, German, Greek, Italian, Latin, Swedish
Horse Shield of Limb Wood; Noted Protector; Similar to Rose; Horse; Fame; Pretty Rose
Girl/Female
Indian, Tamil
Slender Stem of a Creeper
Male
Portuguese
Portuguese form of Latin Philippus, FILIPE means "lover of horses."
Girl/Female
Afghan, Arabic, Indian, Kannada, Muslim
Immortal; Deathless
Boy/Male
Muslim
The servant of the protector
Boy/Male
Indian, Tamil
Combination of Lord Krishna / Shiva
Girl/Female
Indian
One who makes something clear, Manifest, One who clarifies, Plain
Girl/Female
Hindu
Goddess Lakshmi, Born of the ocean
Boy/Male
Australian, Basque, French, Hebrew
God will Multiply; God will Add
COMMUTING MATRICES
COMMUTING MATRICES
COMMUTING MATRICES
COMMUTING MATRICES
COMMUTING MATRICES
n.
The act of committing, or putting in charge, keeping, or trust; consignment; esp., the act of committing to prison.
p. pr. & vb. n.
of Comminute
p. pr. & vb. n.
of Commune
a.
Having the character of larceny; as, a larcenous act; committing larceny.
a.
Engaged in warfare; fighting; combating; serving as a soldier.
p. pr. & vb. n.
of Confute
p. pr. & vb. n.
of Comment
p. pr. & vb. n.
of Commit
n.
The act or process of computing; calculation; reckoning.
p. pr. & vb. n.
of Commove
p. pr. & vb. n.
of Commute
n.
The act or process of confuting; refutation.
n.
A contrivance for computing the revolutions of a wheel; an odometer.
p. pr. & vb. n.
of Commix
n.
The act or process of commenting or criticising; exposition.
v. i.
To make an enumeration or computation; to engage in numbering or computing.
a.
Acting in competition; competing; rival.
p. pr. & vb. n.
of Compete
p. pr. & vb. n.
of Compute
p. pr. & vb. n.
of Combat