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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
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
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
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
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
In mathematics, invariant of square matrices
The characteristic polynomial of a square matrix, whose roots are the eigenvalues, is defined as a determinant with polynomial entries. In geometry
Determinant
Square matrix of cyclically shifted rows
column of the matrix; and possibly with a different direction of shift (which is sometimes called an anti-circulant matrix). The polynomial f ( x ) = c
Circulant_matrix
Polynomials used for interpolation
phenomenon of large oscillation. Lagrange polynomials are also deeply related to matrix diagonalization and matrix functions via Frobenius covariants and
Lagrange_polynomial
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 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 Fejér–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Used for the resultant of two polynomials
In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials that have coefficients in a commutative ring (such as an integral
Sylvester_matrix
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
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
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
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_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
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
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
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
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 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
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
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 matrix where a[i,j]=1/(i+j-1)
that is, as a Gramian matrix for powers of x. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices
Hilbert_matrix
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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) 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
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
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
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
Matrix of partial derivatives of a vector-valued function
vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Characteristic polynomial whose associated linear system is stable
A polynomial is defined to be stable in two different ways. Hurwitz-stable: In the context of a differential equation (continuous-time) the characteristic
Stable_polynomial
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
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
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)
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
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
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
Polynomial sequence
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike
Zernike_polynomials
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
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
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
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
multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = ST(x)S(x) where S is a matrix (possibly
SOS-convexity
Polynomial interpolation using derivative values
integration. Hermite interpolating polynomials are also deeply related to matrix diagonalization, Jordan normal form and matrix functions via generalized Frobenius
Hermite_interpolation
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
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
Multiplicative factor in a mathematical expression
a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without
Coefficient
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
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
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
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
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
mapping theorem for the polynomial functional calculus: Let A be an n × n matrix with eigenvalues λ1, ..., λn, then for any polynomial p, p(A) has eigenvalues
Jordan_normal_form
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
Complexity class used to classify decision problems
the cities. Then verification can clearly be done in polynomial time. It simply adds the matrix entries corresponding to the paths between the cities
NP_(complexity)
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively
Positive_polynomial
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
Topics referred to by the same term
Hurwitz-stable matrix is a matrix whose eigenvalues all have negative real part. The Routh–Hurwitz matrix associated to a polynomial is a particular matrix whose
Hurwitz_matrix
Alexander matrix is a presentation matrix for the Alexander invariant of a knot. The determinant of an Alexander matrix is the Alexander polynomial for the
Alexander_matrix
Mathematical convex optimization
Interior Point Polynomial Methods in Convex Programming. SIAM, 1994. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in
Linear_matrix_inequality
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
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
Mathematical invariant of a knot or link
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant
Jones_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
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
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
Product of pairwise differences
In algebra, the Vandermonde polynomial of an ordered set of n variables X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} , named after Alexandre-Théophile
Vandermonde_polynomial
Canonical form of matrices over a field
The matrix of the linear operator with respect to such a basis is the companion matrix of a monic polynomial; this polynomial (the minimal polynomial of
Frobenius_normal_form
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
Mathematical approximation of a function
of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function
Taylor_series
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)
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
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MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
MATRIX POLYNOMIAL
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