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In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value.
Synonym for (0,1)-matrix, binary matrix or Boolean matrix. Can be used to represent a k-adic relation. Markov matrix: A matrix of non-negative real numbers, such that the entries in each row sum to 1. Metzler matrix: A matrix whose off-diagonal entries are non-negative. Monomial matrix: A square matrix with exactly one non-zero entry in each ...
To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple (,, ()) where is a finite-dimensional complex vector space, is a finite set and () is a family of one-dimensional subspaces of such that =.
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.) Moment matrices play an important role in polynomial fitting , polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares ) [ 1 ] and econometrics .
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In particular, the trace (the sum of the elements of the diagonal) is the value of e 1, and thus the sum of the eigenvalues. Similarly, the determinant is – up to the sign – the constant term of the characteristic polynomial, i.e. the value of e n. Thus the determinant of a square matrix is the product of the eigenvalues.