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Matrices similar to diagonal matrices
linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists
Diagonalizable_matrix
Mathematical operation on invertible matrices
log A {\displaystyle \log A} for a diagonalizable matrix A {\displaystyle A} is the following: Find the matrix V {\displaystyle V} of eigenvectors of
Logarithm_of_a_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 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
Matrix that commutes with its conjugate transpose
there exists a unitary matrix U such that UAU* and UBU* are diagonal matrices. In other words A and B are simultaneously diagonalizable. In this special case
Normal_matrix
Matrix decomposition
{\displaystyle 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
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 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
Specialized notation for multivariable calculus
The matrix function f ( X ) {\displaystyle f(\mathbf {X} )} is defined in terms of the scalar function f ( x ) {\displaystyle f(x)} for diagonalizable matrices
Matrix_calculus
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
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
Mathematical concept in algebra
{\displaystyle AB=BA} ) if they are simultaneously diagonalizable (that is, there exists an invertible matrix P {\displaystyle P} such that both P − 1 A P {\displaystyle
Commuting_matrices
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
Matrix equal to its conjugate-transpose
}}\mathbf {v} \in \mathbb {C} ^{n}.} A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues. Hermitian matrices
Hermitian_matrix
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
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)
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
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 equal to its transpose
general (complex) symmetric matrix may be defective and thus not be diagonalizable. If A {\displaystyle A} is diagonalizable it may be decomposed as A =
Symmetric_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
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
Idempotent linear transformation from a vector space to itself
vector space of dimension d {\displaystyle d} over a field is a diagonalizable matrix, since its minimal polynomial divides x 2 − x {\displaystyle x^{2}-x}
Projection_(linear_algebra)
Square matrix constructed from a monic polynomial
C(p)} is diagonalizable as C ( p ) = V − 1 D V {\displaystyle C(p)=V^{-1}\!DV} , where D is the diagonal matrix and V is the Vandermonde matrix corresponding
Companion_matrix
Matrix of geometric progressions
of configuration space. Companion matrix § Diagonalizability Schur polynomial – a generalization Alternant matrix Lagrange polynomial Wronskian List
Vandermonde_matrix
Complex matrix whose conjugate transpose equals its inverse
∗ {\displaystyle 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 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
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
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
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
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
{\displaystyle L_{i}} contains v. (Since any permutation matrix is normal and any normal matrix is diagonalizable over the complex numbers, the algebraic and geometric
Permutation_matrix
Matrix B such that B² equals a given matrix A
triangular matrix are precisely its eigenvalues (see Triangular matrix#Properties). An n × n matrix A is diagonalizable if there is a matrix V and a diagonal
Square_root_of_a_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
Function that maps matrices to matrices
square matrix A is diagonalizable, if there is an invertible matrix P such that D = P − 1 A P {\displaystyle D=P^{-1}\,A\,P} is a diagonal matrix, that
Analytic_function_of_a_matrix
Relation of a matrix of variables between two points in time
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related
Matrix_difference_equation
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
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)
Generalization of signum function to matrices
n}} , then csgn ( A ) {\displaystyle \operatorname {csgn} (A)} is diagonalizable and has eigenvalues that are ± 1 {\displaystyle \pm 1} . Theorem: Let
Matrix_sign_function
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 = [ 5 4 2 1 0 1
Jordan_normal_form
Matrices important in quantum mechanics and the study of spin
linear factors is diagonalizable) means this implies a → ⋅ σ → {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } is diagonalizable with possible eigenvalues
Pauli_matrices
Formulation of quantum mechanics
{\displaystyle [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
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
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 decomposition
× n {\displaystyle m\times n} matrix, whereas eigenvalue decomposition can only be applied to square diagonalizable matrices. Nevertheless, the two
Singular_value_decomposition
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
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
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 C l 1 , 3 ( R ) . {\displaystyle
Gamma_matrices
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
the reals, if the matrix of an endomorphism is symmetric, then it is diagonalizable. Active and passive transformation Covariance and contravariance of
Change_of_basis
Vector satisfying some of the criteria of an eigenvector
complete basis for V {\displaystyle V} . That is, the matrix A {\displaystyle A} may not be diagonalizable. This happens when the algebraic multiplicity of
Generalized_eigenvector
Matrix that, squared, equals itself
principle of induction, the result follows. An idempotent matrix is always diagonalizable. Its eigenvalues are either 0 or 1: if x {\displaystyle \mathbf
Idempotent_matrix
Type of matrix representation
complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle
Polar_decomposition
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
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
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
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
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
Largest absolute value of an operator's eigenvalues
Hermitian matrix and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and
Spectral_radius
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
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
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
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)
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
Theorem on linear operators
linear combinations T = α A + β B {\displaystyle T=\alpha A+\beta B} are diagonalizable for all α , β ∈ C {\displaystyle \alpha ,\beta \in \mathbb {C} } . Then
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
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
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
} The 2 × 2 matrix is real symmetric and so (by the spectral theorem) it is orthogonally diagonalizable. That is, there is an angle θ
Neutrino_oscillation
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
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
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
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
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)
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
Type of partial differential equations
the matrix A := α 1 A 1 + ⋯ + α d A d {\displaystyle A:=\alpha _{1}A^{1}+\cdots +\alpha _{d}A^{d}} has only real eigenvalues and is diagonalizable. If
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
Square (0,1) matrix
mathematics, a Redheffer matrix, often denoted A n {\displaystyle A_{n}} as studied by Redheffer (1977), is a square (0,1) matrix whose entries aij are 1
Redheffer_matrix
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
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
(on a complex Hilbert space) continuous linear operator
operator on a finite-dimensional inner product space) is unitarily diagonalizable. Let T {\displaystyle T} be a bounded operator. The following are equivalent
Normal_operator
Direct sum of simple Lie algebras
characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over the algebraic closure) and nilpotent part x = s + n {\displaystyle
Semisimple_Lie_algebra
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
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
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
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
Map from a Lie algebra to its Lie group
\mathbb {C} } -diagonalizable matrices with eigenvalues either positive or with modulus 1 {\displaystyle 1} , and of non-diagonalizable matrices with a
Exponential_map_(Lie_theory)
Type of mathematical system
be confirmed by substitution. If A {\displaystyle \mathbf {A} } is diagonalizable, then any vector in an N {\displaystyle N} -dimensional space can be
Linear_dynamical_system
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
Type of group in mathematics
because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix A = [ a b c d ] {\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}
Orthogonal_group
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
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
Mathematical study of linear operators
show that A {\displaystyle A} is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A = U T U ∗ {\displaystyle A=UTU^{\ast
Operator_theory
Mathematical expression for linear operators
operators again potentially diagonalisable (because they are simultaneously diagonalizable over the algebraic closure of K {\displaystyle K} ). Since the only
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Neuroscientific theory
differential equations. Since C {\displaystyle C} is symmetric, it is also diagonalizable, and the solution can be found, by working in its eigenvectors basis
Hebbian_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
Quantum mechanical spectroscopic effect
, when is arises from dipole–dipole interaction) matrix, which means that it is diagonalizable. With D j j {\displaystyle D_{jj}} denoted as D j {\displaystyle
Zero-field_splitting
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)
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
diagonalization of a finite set of matrices under the assumption that each matrix is diagonalizable over the field of the complex numbers. A wild problem. sometimes
Wild_problem
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
Method used in statistics, pattern recognition, and other fields
eigenvalue. If Σ − 1 Σ b {\displaystyle \Sigma ^{-1}\Sigma _{b}} is diagonalizable, the variability between features will be contained in the subspace
Linear_discriminant_analysis
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DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX