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The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix. The rank of the Gram matrix of vectors in or equals the dimension of the space ...
The Gram matrix of a sequence of points ,, …, in k-dimensional space ℝ k is the n×n matrix = of their dot products (here a point is thought of as a vector from 0 to that point):
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1. A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. [ 1 ] Kernel methods (for instance, support vector machines or Gaussian processes [ 2 ] ) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane.
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization).
The reason may be, at least partly, price. Toledano declined to disclose how much the fragment used for the B/1M cost, but he noted that raw meteorite can sell for more, per gram, than gold.
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2 n−1 times the maximal determinant of a {0,1} matrix of size n−1.