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Matrix rings are non-commutative and have no unique factorization: there are, in general, many ways of writing a matrix as a product of matrices. Thus, the factorization problem consists of finding factors of specified types. For example, the LU decomposition gives a matrix as the product of a lower triangular matrix by an upper triangular matrix.
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
Microsoft Math in Bing app – Math helper as a feature within the Bing mobile app on iOS and Android platforms, released in August 2018 [12] Microsoft Math Solver – Mobile app for iOS (first released in November 2019-No longer available in August 2024.) [13] and Android (first released in December 2019), [14] as well as a Microsoft Edge ...
A matrix M is idempotent when M 2 = M. Idempotent matrices generalize the idempotent properties of 0 and 1. Idempotent matrices generalize the idempotent properties of 0 and 1. The completion of the square method of addressing the equation a 2 + b 2 = a , {\displaystyle a^{2}+b^{2}=a,} shows that some idempotent 2×2 matrices are parametrized ...
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as AB = pI , where A and B are square matrices and I is the identity matrix . [ 1 ]