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Matrices similar to diagonal matrices
square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a
Diagonalizable_matrix
Property of a mathematical matrix
contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation
Definite_matrix
Mathematical operation on invertible matrices
rotation matrices. A method for finding logA for a diagonalizable matrix A is the following: Find the matrix V of eigenvectors of A (each column of V is an
Logarithm_of_a_matrix
Matrix decomposition
of A. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Matrix that commutes with its conjugate transpose
of a normal matrix. Let A be a n × n complex matrix. Then the following are equivalent: A is normal. A is diagonalizable by a unitary matrix. There exists
Normal_matrix
Non-diagonalizable matrix; one lacking a basis of eigenvectors
algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an
Defective_matrix
Matrix with a multiplicative inverse
diagonalization. An arbitrary matrix 𝐀 is diagonalizable if and only if 𝐀=𝐏𝐃𝐏−1 where 𝐃 is a diagonal matrix and 𝐏 is an invertible matrix. This expression
Invertible_matrix
Matrix operation generalizing exponentiation of scalar numbers
P is the minimal polynomial of A. We further assume that A is a diagonalizable matrix. In particular, the roots of P are simple, and the "interpolation"
Matrix_exponential
Array of numbers
where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable. More generally, and
Matrix_(mathematics)
Mathematical concept in algebra
triangularizable. Two diagonalizable matrices A and B commute (AB=BA) if they are simultaneously diagonalizable (that is, there exists an invertible matrix P such that
Commuting_matrices
Square matrix constructed from a monic polynomial
eigenvalues λ1,…,λn, then C(p) is diagonalizable as C(p)=V−1DV, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ's:
Companion_matrix
Matrix equal to its conjugate-transpose
An n×n matrix A is Hermitian if and only if ⟨𝐯,A𝐯⟩∈ℝ,for all 𝐯∈ℂn. A square matrix is Hermitian if and only if it is unitarily diagonalizable with real
Hermitian_matrix
Matrix whose only nonzero elements are on its main diagonal
diagonal entries of a matrix. Anti-diagonal matrix Banded matrix Bidiagonal matrix Diagonally dominant matrix Diagonalizable matrix Jordan normal form Multiplication
Diagonal_matrix
Mathematical theorem
a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a
Bauer–Fike_theorem
Eigenvalue algorithm
known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle
Power_iteration
Algebraic term
quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic
Unipotent
Set of a matrix's eigenvalues
decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in
Spectrum_of_a_matrix
Result about when a matrix can be diagonalized
computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix of eigenvalues. The
Spectral_theorem
Specialized notation for multivariable calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various
Matrix_calculus
Matrix equal to its transpose
symmetric matrix may be defective and thus not be diagonalizable. If A is diagonalizable it may be decomposed as A=QΛQ𝖳 where Q is an orthogonal matrix QQ𝖳=I
Symmetric_matrix
Complex matrix whose conjugate transpose equals its inverse
Uy⟩ = ⟨x, y⟩. U is normal (U∗U=UU∗). U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem
Unitary_matrix
Matrix defined using smaller matrices called blocks
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices
Block_matrix
Idempotent linear transformation from a vector space to itself
projection P=P2 on a vector space of dimension d over a field is a diagonalizable matrix, since its minimal polynomial divides x2−x, which splits into distinct
Projection_(linear_algebra)
Mathematical concept in algebra
nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix. See also:
Nilpotent_matrix
Block diagonal matrix of Jordan blocks
diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all 1 × 1. More
Jordan_matrix
Matrix of inner products of vectors
Gram matrix over the reals is a symmetric matrix, it is diagonalizable and its eigenvalues are non-negative. The diagonalization of the Gram matrix is the
Gram_matrix
Equivalence under a change of basis (linear algebra)
matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex
Matrix_similarity
Matrix of geometric progressions
of configuration space. Companion matrix § Diagonalizability Schur polynomial – a generalization Alternant matrix Lagrange polynomial Wronskian List
Vandermonde_matrix
Function that maps matrices to matrices
corresponding matrix series converges for matrices A such that ‖A‖<r for some matrix norm that satisfies ‖AB‖≤‖A‖‖B‖. A square matrix A is diagonalizable, if there
Analytic_function_of_a_matrix
Vector satisfying some of the criteria of an eigenvector
eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable. This happens when the algebraic multiplicity of at least
Generalized_eigenvector
Matrix decomposition
it can be applied to any m×n matrix, whereas eigenvalue decomposition can only be applied to square diagonalizable matrices. Nevertheless, the two
Singular_value_decomposition
Type of mathematical equation
that the matrix A be diagonalizable and bypasses complexities of the Jordan canonical forms normally utilized. A first-order homogeneous matrix ordinary
Matrix_differential_equation
Matrix with exactly one 1 per row and column
for which Li contains v. (Since any permutation matrix is normal and any normal matrix is diagonalizable over the complex numbers, the algebraic and geometric
Permutation_matrix
Concepts from linear algebra
transformation. Such a matrix A is said to be similar to the diagonal matrix Λ, or diagonalizable. The matrix Q is the change of basis matrix of the similarity
Eigenvalues_and_eigenvectors
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Matrices important in quantum mechanics and the study of spin
(right/left). Each Pauli matrix is Hermitian, and together with the identity matrix 𝕀 (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices
Pauli_matrices
Generalization of signum function to matrices
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts
Matrix_sign_function
Machine learning framework for portfolio construction
number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest
Hierarchical_Risk_Parity
Formula in matrix theory
of A, which are (projection) matrix Lagrange polynomials of A. Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues
Sylvester's_formula
Matrix whose conjugate transpose is its negative (additive inverse)
skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their
Skew-Hermitian_matrix
Square matrices satisfy their characteristic equation
D of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices (for a matrix to be diagonalizable it suffices
Cayley–Hamilton_theorem
Formulation of quantum mechanics
rigor is required, [P,f(X)]=−if′(X). Since X is a Hermitian matrix, it should be diagonalizable, and it will be clear from the eventual form of P that every
Matrix_mechanics
Matrix that, squared, equals itself
idempotent matrix is always diagonalizable. Its eigenvalues are either 0 or 1: if 𝐱 is a non-zero eigenvector of some idempotent matrix A and λ its
Idempotent_matrix
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following matrix: A=[542101−1−1−1−13011−12]
Jordan_normal_form
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable matrix Triangular
Outline_of_linear_algebra
Coordinate change in linear algebra
"new" (other) bases. A is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinates of the "new"
Change_of_basis
Generators of the Clifford algebra for relativistic quantum mechanics
matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(ℝ). It is also possible to
Gamma_matrices
Branch of mathematics
is square-free, then the matrix is diagonalizable. A symmetric matrix is always diagonalizable. There are non-diagonalizable matrices, the simplest being
Linear_algebra
Matrix B such that B² equals a given matrix A
triangular matrix are precisely its eigenvalues (see Properties of a triangular matrix). An n × n matrix A is diagonalizable if there is a matrix V and a
Square_root_of_a_matrix
commuting nilpotent matrix and diagonalizable matrix Schur decomposition — similarity transform bringing the matrix to a triangular matrix Singular value decomposition
List of numerical analysis topics
List_of_numerical_analysis_topics
matrix f(A) as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of A. Let A be a diagonalizable matrix with
Frobenius_covariant
Theorem in linear algebra
Given that M is diagonalizable, M is conjugate to a diagonal matrix with eigenvalues r1, ... , rn on the diagonal (denote r1 = r). The matrix Mk/rk will be
Perron–Frobenius_theorem
Topics referred to by the same term
an abstract element of an algebraic structure that generalizes a diagonalizable matrix. A precise meaning depends on context: A semisimple element in the
Semisimple_element
Type of matrix representation
or complex matrix A is a factorization of the form A=UP, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal
Polar_decomposition
Principle in geometry and linear algebra
The matrix A in the above decomposition is a symmetric matrix. In particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by
Principal_axis_theorem
Largest absolute value of an operator's eigenvalues
when A is a Hermitian matrix and ‖⋅‖ is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices
Spectral_radius
A matrix canonical form
linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices
Weyr_canonical_form
Polynomial associated with a matrix
endomorphism φ of a finite-dimensional vector space over a field F is diagonalizable if and only if its minimal polynomial factors completely over F into
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
Relation of a matrix of variables between two points in time
is diagonalizable, we can rewrite A in terms of its eigenvalues and eigenvectors, giving the solution as 𝐲t=𝐏𝐃t𝐏−1𝐲0, where P is an n × n matrix whose
Matrix_difference_equation
Concept in Lie algebra representation theory
the algebra, i.e., to there being simultaneously diagonalizable matrices (see diagonalizable matrix). If G is a Lie group with Lie algebra g {\displaystyle
Weight (representation theory)
Weight_(representation_theory)
Linear algebra aspects of graph theory
matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable;
Spectral_graph_theory
Numerical methods for matrix eigenvalue calculation
and Fike proved that if λ is an eigenvalue for a diagonalizable n × n matrix A with eigenvector matrix V, then the absolute error in calculating λ is bounded
Eigenvalue_algorithm
Canonical form of matrices over a field
the respective actions on these subspaces. For instance if both are diagonalizable, then one can take the decomposition into eigenspaces (for which the
Frobenius_normal_form
Construct for Hermitian matrices
Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches
Rayleigh_quotient
Theorem on linear operators
Furthermore, let A,B∈B(X) be such that all linear combinations T=αA+βB are diagonalizable for all α,β∈ℂ. Then all eigenvalues of T are of the form λT=αλA+βλB
Motzkin–Taussky_theorem
Isomorphism of projective spaces in geometry
× (n+1) matrix that has an eigenspace of dimension n. It is a homology, if the matrix has another eigenvalue and is therefore diagonalizable. It is an
Homography
Branch of mathematical analysis
root of a matrix, matrix exponential, and logarithm of a matrix are basic examples of hypercomplex analysis. The function theory of diagonalizable matrices
Hypercomplex_analysis
Concept in physics
pseudo-Hermiticity regardless of diagonalizability, thereby applying to the physically interesting case of non-diagonalizable Hamiltonians at exceptional points
Non-Hermitian quantum mechanics
Non-Hermitian_quantum_mechanics
Type of partial differential equations
real eigenvalues and is diagonalizable. If the matrix A has s distinct real eigenvalues, it follows that it is diagonalizable. In this case the system
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
Mathematical property
semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These
Semi-simplicity
Phenomenon in which a neutrino changes lepton flavor as it travels
real symmetric and so (by the spectral theorem) it is orthogonally diagonalizable. That is, there is an angle θ such that if we define
Neutrino_oscillation
Random process independent of past history
distribution π is a left eigenvector of row stochastic matrix P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors
Markov_chain
On when a definite intersection form of a smooth 4-manifold is diagonalizable
is diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over
Donaldson's_theorem
Direct sum of simple Lie algebras
h∈𝔥, ad(h) is diagonalizable. As it turns out, 𝔥 is abelian and so all the operators in ad(𝔥) are simultaneously diagonalizable. For each linear
Semisimple_Lie_algebra
Theorem
sum is n, so they must all be 1. The matrix P is Hermitian, therefore diagonalizable, so it is the identity matrix—in other words the columns of M are
Hadamard's_inequality
Statistical theorem in the analysis of variance
simultaneously diagonalizable in this vector space (and hence also together with B(1)). By iteration it follows that all B-s are simultaneously diagonalizable. Thus
Cochran's_theorem
field K. If the order of g is coprime to p then g is diagonalizable and represented by a diagonal matrix diag(1, ... , 1, r ) = [ 1 0 0 ⋯ 0 0 1 0 ⋯ 0 ⋮ ⋮
Pseudoreflection
Belgian mathematician (1921–1999)
version only applied to systems that could be represented by a diagonalizable matrix. The test was subsequently generalised by Vasile M. Popov (in 1966)
Vitold_Belevitch
Type of mathematical system
equation is 𝐱(t)=𝐫keλkt as may be confirmed by substitution. If 𝐀 is diagonalizable, then any vector in an N-dimensional space can be represented by a linear
Linear_dynamical_system
Functional analysis concept
form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Operators useful in quantum mechanics
B and C have a common set of eigenfunctions (and are simultaneously diagonalizable), whereas p and q famously don't and aren't. Feynman 1998, p. 151 Dirac
Creation and annihilation operators
Creation_and_annihilation_operators
Function that is its own inverse
characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s on the diagonal of the corresponding matrix. If the operator is orthogonal
Involution_(mathematics)
Square (0,1) matrix
mathematics, a Redheffer matrix, named after Raymond Redheffer and often denoted A n , {\displaystyle A_{n},} is a square (0,1) matrix whose entries aij are
Redheffer_matrix
hyperbolic system at each cell interface. The Roe matrix must obey the following conditions: Diagonalizable with real eigenvalues: ensures that the new linear
Roe_solver
(on a complex Hilbert space) continuous linear operator
operator on a finite-dimensional inner product space) is unitarily diagonalizable. Let T be a bounded operator. The following are equivalent. T is normal
Normal_operator
Nilpotent subalgebra of a Lie algebra
\operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}} is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition
Cartan_subalgebra
Number of positive, negative and zero eigenvalues of a metric tensor
independent. By the spectral theorem a symmetric n × n matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted
Metric_signature
Type of group in mathematics
that the matrix of the quadratic form is Q=[100−ω], because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix A=[abcd]
Orthogonal_group
Map from a Lie algebra to its Lie group
Its image consists of ℂ-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated
Exponential_map_(Lie_theory)
Adjustment of color intensities in photography
sum of N fixed linear transformations, not necessarily consistently diagonalizable. Color cast Color temperature Gamma correction White point Phyllis Davis
Color_balance
A is an n×n self-adjoint matrix, there is an orthonormal basis of eigenvectors of A. This implies that A is diagonalizable. Furthermore, each eigenvalue
Transform_theory
Quantum mechanical spectroscopic effect
(Dxx+Dyy+Dzz=0, when is arises from dipole–dipole interaction) matrix, which means that it is diagonalizable. With Djj denoted as Dj for simplicity, the Hamiltonian
Zero-field_splitting
Neuroscientific theory
coupled linear differential equations. Since C is symmetric, it is also diagonalizable, and the solution can be found, by working in its eigenvectors basis
Hebbian_theory
Mathematical study of linear operators
A∗A=AA∗. One can show that A is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A=UTU∗, where U is unitary and
Operator_theory
Theory in supersymmetric gauge theory
{\displaystyle \phi } is traceless and diagonalizable so can be gauge rotated to (is in the conjugacy class of) a matrix of the form 1 2 a σ 3 {\displaystyle
Seiberg–Witten_theory
Representation of the symmetry group of spacetime in special relativity
section 2.1 : This can be seen as follows. The matrix q has eigenvalues {−1, −1}, but it is not diagonalizable. If q = exp(Q), then Q has eigenvalues λ, −λ
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Measure of the ability of a porous material to allow fluids to pass through it
pressure gradient) is always positive The permeability tensor is always diagonalizable (being both symmetric and positive definite). The eigenvectors will
Permeability_(porous_media)
Product of the principal curvatures of a surface
symmetric matrix of second derivatives, which in Euclidean space is diagonalizable via the special case of the spectral theorem restricted to when H is
Gaussian_curvature
Mathematical expression for linear operators
potentially diagonalisable respectively (because they are simultaneously diagonalizable over the algebraic closure of K {\displaystyle K} ). Since the only
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Universal construction of a complex Lie group from a real Lie group
acting on a finite-dimensional vector space W with operators in 𝖆 diagonalizable and operators in 𝖓 nilpotent, there is a vector w that is an eigenvector
Complexification_(Lie_group)
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DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
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