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The locations of the points in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution from the distribution , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}
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.
Commutation matrix: The matrix of the linear map that maps a matrix to its transpose: See Vectorization: Duplication matrix: The matrix of the linear map mapping the vector of the distinct entries of a symmetric matrix to the vector of all entries of the matrix: See Vectorization: Elimination matrix
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 ...
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.
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 other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
When p = ±3, the above values of t 0 are sometimes called the Chebyshev cube root. [29] More precisely, the values involving cosines and hyperbolic cosines define, when p = −3 , the same analytic function denoted C 1/3 ( q ) , which is the proper Chebyshev cube root.