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A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.
PL/I provides two facilities for array slicing. Using iSub DEFINING, an array slice can be declared using iSUB variables to map specific elements in a "base array" onto elements of the "defined array". iSUBs can define rows, columns, diagonals, or many-to-one mappings.
Elimination matrix: The matrix of the linear map mapping the vector of the entries of a matrix to the vector of a part of the entries (for example the vector of the entries that are not below the main diagonal) See vectorization: Exchange matrix: The binary matrix with ones on the anti-diagonal, and zeroes everywhere else. a ij = δ n+1−i,j
Functions of a single variable (such as sine and cosine) may be implemented by a simple array. Functions involving two or more variables require multidimensional array indexing techniques. The latter case may thus employ a two-dimensional array of power[x][y] to replace a function to calculate x y for a limited range of x and y values ...
It is called an identity matrix because multiplication with it leaves a matrix unchanged: = = for any m-by-n matrix A. A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group ...
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
A bilinear map is a function: such that for all , the map (,) is a linear map from to , and for all , the map (,) is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.