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In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. [ 1 ]
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs.
Synonym for binary matrix or logical matrix. Alternant matrix: A matrix in which successive columns have a particular function applied to their entries. Alternating sign matrix: A square matrix with entries 0, 1 and −1 such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. Anti-diagonal ...
The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix. Properties. If any component of an alternating ...
A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(K) ≠ 2).
Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .; The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations.It is a popular method for solving the large matrix equations that arise in systems theory and control, [1] and can be formulated to construct solutions in a memory-efficient, factored form.
Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that () = =, while, by convention, 0-forms are defined to be scalars: () = =. The determinant on n × n {\displaystyle n\times n} matrices, viewed as an n {\displaystyle n} argument function of the column vectors, is an important example of an alternating ...