In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that...

Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial...

integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful within number...

linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as...

mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers...

test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable...

square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity of the original polynomial. The greatest...

eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors...

the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p = p 1 p 2 2 ⋯ p k k {\displaystyle p=p_{1}p_{2}^{2}\cdots...

minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with...

situation is better than for integer factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most...

mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition...

roots of a polynomial determinant. Matrix theory is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear...

semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in Positive definite matrix § Decomposition...

Its factorization was announced on April 1, 1991, by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple-polynomial quadratic...

means that the Vandermonde matrix is the design matrix of polynomial regression. In numerical analysis, solving the equation V a = y {\displaystyle Va=y}...

constant α > 0 {\displaystyle \alpha >0} is a polynomial time algorithm. The following table summarizes some classes of commonly encountered time complexities...

methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose...

Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shifted QR-factorization. In eigenvalue...

Furthermore, a similar factorization holds for any n × n rotation matrix. If the dimension, n, is odd, there will be a "dangling" eigenvalue of 1; and for any...

Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra...

invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} , such that: T ( x ) = A x {\displaystyle T(\mathbf {x} )=A\mathbf...

Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular...

yields a zero remainder, indicating that r1(x) is the greatest common divisor polynomial of a(x) and b(x), consistent with their factorization. Many of the...

they evaluate to zero. Factorization consists of rewriting a polynomial as a product of several factors. For example, the polynomial x 2 − 3 x − 10 {\displaystyle...

symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial...

factorization, introduced by George David Birkhoff at 1909, is the presentation of an invertible matrix with polynomial coefficients as a product of three...

of K {\displaystyle K} is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated to x is a monic polynomial with...

power of p. More generally, if a polynomial factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo...