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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
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
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
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
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
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
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
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
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
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
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)
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
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 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
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
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)
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
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 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
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
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
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
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 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
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 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
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
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable matrix Triangular
Outline_of_linear_algebra
Matrix of geometric progressions
of configuration space. Companion matrix § Diagonalizability Schur polynomial – a generalization Alternant matrix Lagrange polynomial Wronskian List
Vandermonde_matrix
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
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
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)
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
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
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
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
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
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
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
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
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
Mathematical operation
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
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
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
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
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 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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
(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
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
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
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
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
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 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
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
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
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
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
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
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)
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
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
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
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
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)
Converting classical mechanics to quantum mechanics
and accurately described by a single wave function single system-wide diagonalizable state. From this perspective, first quantization is not a truly multi-particle
First_quantization
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)
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
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
Evaluates how likely it is that any difference between data sets arose by chance
symmetric ( m − 1 ) × ( m − 1 ) {\displaystyle (m-1)\times (m-1)} matrix, and hence diagonalizable. It is therefore possible to make a linear change of variables
Pearson's_chi-squared_test
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
Girl/Female
Indian, Punjabi, Sikh
Deities of Fortune
Boy/Male
British, English
Storekeeper
Boy/Male
Anglo, Australian, British, English
Of Glorious Valor
Boy/Male
Australian, Chinese
Thunder; Upright
Female
Spanish
 Pet form of Spanish MarÃa, MARITA means "obstinacy, rebelliousness" or "their rebellion."
Male
French
 Old French form of German Lanzo, LANCE means "land." Compare with another form of Lance.
Girl/Female
Hindu
Sacred plant, A sacred plant (Basil)
Girl/Female
Gujarati, Hindu, Indian, Sanskrit
Beautiful Deer
Girl/Female
Gujarati, Hindu, Indian, Mythological, Tamil
The One who Blesses
Female
Egyptian
, the wife of Seti-nekht.
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
DIAGONALIZABLE MATRIX
n.
In type founding and forging, an impression or matrix, formed by a punch drift.
n.
A mold or matrix in which anything is cast or formed to a particular shape.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
See Matrix.
v. i.
The mineral substance which incloses a vein; a matrix; a gangue.
n.
The womb.
n.
A kind of cartilage with a fibrous matrix and approaching fibrous connective tissue in structure.
n.
The amorphous or homogenous matrix or ground mass, as distinguished from well-defined crystals; as, the magma of porphyry.
v. t.
The white fibrous matter forming the matrix from which fungi.
n.
A mold; a matrix.
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.
n.
The matrix, or cavity, in which anything is shaped, and from which it takes its form; also, the body or mass containing the cavity; as, a sand mold; a jelly mold.
n.
A protoplasmic animal cell; esp., such as float free, like blood, lymph, and pus corpuscles; or such as are imbedded in an intercellular matrix, like connective tissue and cartilage corpuscles. See Blood.
n.
One of the protoplasmic cells which occur in the osteogenetic layer of the periosteum, and from or around which the matrix of the bone is developed; an osteoplast.
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
n.
A cutting or engraving; a figure cut into something, as a gem, so as to make a design depressed below the surface of the material; hence, anything so carved or impressed, as a gem, matrix, etc.; -- opposed to cameo. Also used adjectively.
pl.
of Matrix
n.
Hence, that which gives form or origin to anything