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If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of A + uv T, where u and v are column vectors. Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation ...
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.
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns). The multilinear polynomials in n {\displaystyle n} variables form a 2 n {\displaystyle 2^{n}} -dimensional vector space , which is also the basis used in the ...
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map: that is separately -linear in each of its arguments. [1] More generally, one can define multilinear forms on a module over a commutative ring.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
In the above moment-cumulant formula\ = (:) for joint cumulants, one sums over all partitions of the set { 1, ..., n}. If instead, one sums only over the noncrossing partitions , then, by solving these formulae for the κ {\textstyle \kappa } in terms of the moments, one gets free cumulants rather than conventional cumulants treated above.
The proof below is based on formal manipulations only, and avoids using any particular interpretation of determinants, which may be taken to be defined by the Leibniz formula. Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal rows or columns) are used; in particular ...