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Consider the system of equations + + = + + = + + = The coefficient matrix is = [], and the augmented matrix is (|) = []. Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions.
An n × n matrix A is said to be skew-centrosymmetric if its entries satisfy , = +, +, {, …,}. Equivalently, A is skew-centrosymmetric if AJ = −JA, where J is the exchange matrix defined previously.
A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation.Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts.For example, a 2,1 represents the element at the second row and first column of the matrix.
It is used to model and optimize the generation, transmission, and distribution of electric power. Linear algebraic concepts such as matrix operations and eigenvalue problems are employed to enhance the efficiency, reliability, and economic performance of power systems .
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.
That method is commonly used for analyzing and clustering textual data and is also related to the latent class model. NMF with the least-squares objective is equivalent to a relaxed form of K-means clustering : the matrix factor W contains cluster centroids and H contains cluster membership indicators.