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Polynomial with a matrix as variable
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial P ( x ) = ∑ i = 0 n
Matrix_polynomial
Matrix whose entries are polynomials
polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is
Polynomial_matrix
Polynomial whose roots are the eigenvalues of a matrix
linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as
Characteristic_polynomial
Type of mathematical expression
identity matrix. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial
Polynomial
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
Polynomial associated with a matrix
linear algebra, the minimal polynomial μA of an n × n {\displaystyle n\times n} matrix A over a field F is the monic polynomial μA over F of least degree
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
Square matrices satisfy their characteristic equation
own characteristic equation. The characteristic polynomial of an n × n {\displaystyle n\times n} matrix A is defined as p A ( λ ) = det ( λ I n − A ) {\displaystyle
Cayley–Hamilton_theorem
Matrix of geometric progressions
theorem for polynomials. In statistics, the equation V a = y {\displaystyle Va=y} means that the Vandermonde matrix is the design matrix of polynomial regression
Vandermonde_matrix
Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Mathematical algorithm
the characteristic polynomial p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda I_{n}-A)} of a square matrix, A, named after Dmitry
Faddeev–LeVerrier_algorithm
Linear algebra matrix
x^{n}-1)} . Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix P {\displaystyle P} : C = c 0 I
Circulant_matrix
In mathematics, invariant of square matrices
efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional
Determinant
Algorithms for polynomial evaluation
enables to involve fast matrix multiplication algorithms to evaluate a polynomial in a series of points. Arbitrary polynomials can be evaluated with fewer
Polynomial_evaluation
Two-dimensional matrix barcode
A Data Matrix is a two-dimensional code consisting of black and white "cells" or dots arranged in either a square or rectangular pattern, also known as
Data_Matrix
Concepts from linear algebra
polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some companion matrix of
Eigenvalues_and_eigenvectors
For a square matrix, the transpose of the cofactor matrix
n × n matrix has entries over a field with at least 2n + 1 elements (e.g. a 5 × 5 matrix over the integers modulo 11). det(A+tI) is a polynomial in t with
Adjugate_matrix
Matrix representing a Euclidean rotation
characteristic polynomial will have degree n and thus n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be
Rotation_matrix
Used for the resultant of two polynomials
matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of
Sylvester_matrix
On the number of spanning trees in a graph
graph's Laplacian matrix. This shows in particular that the number of spanning trees can be computed from the graph data in polynomial time. Kirchhoff's
Kirchhoff's_theorem
Matrix operation generalizing exponentiation of scalar numbers
Cayley–Hamilton theorem the matrix exponential is expressible as a polynomial of order n−1. If P and Qt are nonzero polynomials in one variable, such that
Matrix_exponential
Function of the coefficients of a polynomial that gives information on its roots
blocks of the Sylvester matrix is empty). There is no common convention for the discriminant of a constant polynomial (i.e., polynomial of degree 0). For small
Discriminant
Matrix decomposition
the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Mathematical technique
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that
Matrix factorization of a polynomial
Matrix_factorization_of_a_polynomial
Equivalence under a change of basis (linear algebra)
polynomials, of the matrix (with polynomial entries) XIn − A (the same one whose determinant defines the characteristic polynomial). Note that this Smith
Matrix_similarity
Square polynomial matrix in mathematics
unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular
Unimodular_polynomial_matrix
Matrices similar to diagonal matrices
its characteristic polynomial has n {\displaystyle n} distinct roots in F {\displaystyle F} . Let A {\displaystyle A} be a matrix over F {\displaystyle
Diagonalizable_matrix
Mathematical test in control system theory
arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence
Routh–Hurwitz stability criterion
Routh–Hurwitz_stability_criterion
operator or a matrix A evaluates to zero, i.e., is such that P(A) = 0. Note that all characteristic polynomials and minimal polynomials of A are annihilating
Annihilating_polynomial
Estimate of time taken for running an algorithm
Quasi-polynomial time algorithms are algorithms whose running time exhibits quasi-polynomial growth, a type of behavior that may be slower than polynomial time
Time_complexity
Array of numbers
matrix are the roots of its characteristic polynomial, det ( λ I − A ) {\displaystyle \det(\lambda I-A)} . Matrix theory is the branch of mathematics that
Matrix_(mathematics)
Polynomials used for interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a
Lagrange_polynomial
In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Fi(A) associated with the eigenvalues
Frobenius_covariant
roots of a polynomial is to find the eigenvalues of the companion matrix of monic polynomial, which coincides with the roots of the polynomial. There are
Polynomial_root-finding
Special kind of square matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal
Triangular_matrix
Matrix whose determinant is a resultant
In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester
Bézout_matrix
Knot invariant
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander
Alexander_polynomial
Polynomial of the elements of a matrix
permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries
Permanent_(mathematics)
Matrix of partial derivatives of a vector-valued function
inverse is a polynomial function. If f : Rn → Rm is a differentiable function, a critical point of f is a point where the rank of the Jacobian matrix is not
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Greatest common divisor of polynomials
GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Statistics concept
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable
Polynomial_regression
Form of interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through
Polynomial_interpolation
Concept in linear algebra
{\displaystyle K} ) is called a matrix pencil. An important special case arises when P {\displaystyle P} is polynomial: let ℓ ≥ 0 {\displaystyle \ell \geq
Matrix_pencil
Square matrix in which each ascending skew-diagonal from left to right is constant
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal
Hankel_matrix
Complexity class used to classify decision problems
computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is
NP_(complexity)
Mathematical expression
Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes
Newton_polynomial
Matrix with the same number of rows and columns
mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n {\displaystyle
Square_matrix
Mathematical concept in algebra
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k}
Nilpotent_matrix
Mathematical concept in polynomial theory
resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root
Resultant
Most widely known generalized inverse of a matrix
{\displaystyle A^{*}A} is normal and, as a consequence, an EP matrix. One can then find a polynomial p ( t ) {\displaystyle p(t)} such that ( A ∗ A ) + = p (
Moore–Penrose_inverse
Numerical methods for matrix eigenvalue calculation
ten algorithms of 20th century. Any monic polynomial is the characteristic polynomial of its companion matrix. Therefore, a general algorithm for finding
Eigenvalue_algorithm
Matrix operation which flips a matrix over its diagonal
is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose, since they share the same characteristic polynomial. This can also
Transpose
Polynomial sequence
nonlinear operations on Gaussian noise. random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though
Hermite_polynomials
non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.) The Tutte matrix is named after
Tutte_matrix
Central object in linear algebra; mapping vectors to vectors
there exists an m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} , such that: T ( x )
Transformation_matrix
Mathematical function
elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Square matrix used to represent a graph or network
computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices
Adjacency_matrix
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively
Positive_polynomial
Type of symmetric polynomials in mathematics
_{n}}&\dots &x_{n}^{\lambda _{n}}\end{matrix}}\right]} are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes
Schur_polynomial
Matrix of second derivatives
inflection points, since the Hessian determinant is a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite. Refining
Hessian_matrix
Error-correcting codes
technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, broadcast systems
Reed–Solomon_error_correction
Algorithm for polynomial evaluation
computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. It is named after William George Horner, although it is much
Horner's_method
Method to solve optimization problems
polynomial-time algorithm? Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution? Does LP admit a polynomial-time
Linear_programming
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 in discrete mathematics
{1}{\sqrt {N}}}e^{-{\frac {i2\pi }{N}}(m-1)(n-1)}.} This matrix satisfies the matrix polynomial equation: U 4 = I . {\displaystyle \mathbf {U} ^{4}=\mathbf
Discrete_Fourier_transform
Matrix with a multiplicative inverse
algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it
Invertible_matrix
(Mathematical) decomposition into a product
example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful
Factorization
Expression for sums of powers
inverting a matrix easily obtained from the triangle of Pascal. The term Faulhaber polynomials is used by some authors to refer to another polynomial sequence
Faulhaber's_formula
Study of matrices and their algebraic properties
Orthogonal matrix, unitary matrix Symmetric matrix, antisymmetric matrix Stochastic matrix Matrix polynomial Matrix exponential Some authors, e.g. Horn and
Matrix_analysis
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Algorithmic runtime requirements for common math procedures
(December 2021). "Deterministic computation of the characteristic polynomial in the time of matrix multiplication". Journal of Complexity. 67. arXiv:2010.04662
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Square root of the determinant of a skew-symmetric square matrix
of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that
Pfaffian
Function that maps matrices to matrices
{tr(A)}{2}}I-A\right)f'\left({\frac {tr(A)}{2}}\right).} Matrix polynomial Matrix root Matrix logarithm Matrix exponential Matrix sign function Using the semidefinite ordering
Analytic_function_of_a_matrix
In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ
Hamiltonian_cycle_polynomial
Matrix that commutes with its conjugate transpose
Hermitian matrix Least-squares normal matrix Proof: When A {\displaystyle A} is normal, use Lagrange's interpolation formula to construct a polynomial P {\displaystyle
Normal_matrix
Type of polynomial
In algebra, a multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily
Multilinear_polynomial
Polynomial sequence
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series
Bernoulli_polynomials
(k+1)\times 1} . Design matrix Gramian matrix Projection matrix Lasserre, Jean-Bernard, 1953- (2010). Moments, positive polynomials and their applications
Moment_matrix
Property of a mathematical matrix
Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a
Definite_matrix
Matrix with shifting rows
multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent
Toeplitz_matrix
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the
Jordan_normal_form
Group of unitary complex matrices with determinant of 1
by a traceless 3 × 3 Hermitian matrix H, normalized as tr(H2) = 2, can be expressed as a second order matrix polynomial in H: exp ( i θ H ) = [ − 1 3
Special_unitary_group
Method of representing a random variable
Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms
Polynomial_chaos
Characteristic polynomial whose associated linear system is stable
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: all its
Stable_polynomial
Sum of elements on the main diagonal
definition of the characteristic polynomial. If a is a square matrix with small entries and I denotes the identity matrix, then we have approximately det
Trace_(linear_algebra)
Square matrix where a[i,j]=1/(i+j-1)
{n}}\right)} . The method of moments applied to polynomial distributions results in a Hankel matrix, which in the special case of approximating a probability
Hilbert_matrix
Matrix used to analyze the stability of a polynomial by its coefficients
mathematics, the Routh–Hurwitz matrix, or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are
Routh–Hurwitz_matrix
Kind of square matrix in linear algebra
algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries
Hessenberg_matrix
Index of articles associated with the same name
Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values
Graph_polynomial
In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only
Polynomial_SOS
Mathematical operation in linear algebra
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number
Matrix_multiplication
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Computer science concept
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that
Polynomial_hierarchy
Matrix whose only nonzero elements are on its main diagonal
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices
Diagonal_matrix
Form of a matrix
linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the
Skew-symmetric_matrix
Topics referred to by the same term
square matrix or related classes of matrix Routh–Hurwitz matrix, a square matrix constructed with coefficients of a real polynomial Parity-check matrix is
H-matrix
Age-structured model of population growth
growth at rate λ {\displaystyle \lambda } . The characteristic polynomial of the matrix is given by the Euler–Lotka equation. The Leslie model is very
Leslie_matrix
Set of polynomials where any two are orthogonal to each other
orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The
Orthogonal_polynomials
Tool used in probabilistic polynomial identity testing
probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial
Schwartz–Zippel_lemma
Matrices important in quantum mechanics and the study of spin
B.; Zachos, C.K. (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10: 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10
Pauli_matrices
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
Male
Hungarian
Czech and Hungarian form of Greek Patrikios, PATRIK means "patrician, of noble descent."
Female
English
French form of Latin Maria, MARIE means "obstinacy, rebelliousness" or "their rebellion."
Female
Finnish
Finnish form of Greek Margarites, MAARIT means "pearl."
Female
English
English form of Latin Viatrix, BEATRIX means "voyager (through life)."
Female
German
Pet form of German Katarine, KATRIN means "pure."
Male
French
 French form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Male
French
French and German form of Greek Mattathias, MATHIS means "gift of God."
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Male
English
 English form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Male
Italian
Italian form of Hebrew Mattithyah, MATTIA means "gift of God."
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Male
English
Pet form of English Martin, MARTIE means "of/like Mars."
Male
English
Anglicized form of Irish Gaelic MainchÃn, MANNIX means "little monk."
Female
Finnish
Finnish form of Greek Maria, MAARIA means "obstinacy, rebelliousness" or "their rebellion."Â
Girl/Female
Arabic, Australian, Basque, French, Latin
Lady; Feminine of Martin; Warlike
Girl/Female
Biblical
Rain, prison.
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Female
Finnish
Pet form of Finnish Katariina, KATRI means "pure."
Girl/Female
Maori
The Maori form of April.
Female
Welsh
Welsh form of Old French Caterine, CATRIN means "pure."
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
Boy/Male
African, American, British, Christian, Danish, Dutch, English, French, German, Greek, Hindu, Indian, Jamaican, Kannada, Latin, Polish, Romanian, Shakespearean, Swedish, Swiss, Tamil
Dark One; Rich
Girl/Female
Gujarati, Hindu, Indian, Sanskrit, Sindhi, Tamil
Gold; Goddess Parvati
Boy/Male
Arabic, Muslim
Servant of the Gatherer
Girl/Female
Tamil
Harshitha | ஹரà¯à®·à®¿à®¤à®¾
Happy, Full of Joy
Boy/Male
Muslim
Help of the religion (Islam)
Girl/Female
Indian
Name of Rasi
Girl/Female
Arabic, Muslim
Italian Culture; Derived from Sabine
Boy/Male
British, Celtic, English, Greek, Latin
Celtic Form of Ambrose
Boy/Male
Hindu, Indian
Lord Krishna
Female
Spanish
From the name of a town in western Spain, from Arabic wÄdÄ« al-lubb, GUADALUPE means "river of the wolf."
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
See Matrix.
v. t.
The white fibrous matter forming the matrix from which fungi.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.
n.
The martin.
v. i.
The mineral substance which incloses a vein; a matrix; a gangue.
n.
A mold; a matrix.
pl.
of Maori
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
n.
In type founding and forging, an impression or matrix, formed by a punch drift.
n.
A genus of swallows including the purple martin. See Martin.
a.
Of or pertaining to the meter as a standard of measurement; of or pertaining to the decimal system of measurement of which a meter is the unit; as, the metric system; a metric measurement.
pl.
of Matrix
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
Hence, that which gives form or origin to anything
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.
The womb.
a.
Of or pertaining to the Maoris or to their language.
n.
A housekeeper; esp., a woman who manages the domestic economy of a public instution; a head nurse in a hospital; as, the matron of a school or hospital.