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The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
Matrices that can be decomposed as , that is, Gram matrices of some sequence of vectors (columns of ), are well understood — these are precisely positive semidefinite matrices. To relate the Euclidean distance matrix to the Gram matrix, observe that
In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector , where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" is only occasionally used due to the possible confusion with positive-definite matrices, which are different.
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.
In his introduction as North Carolina football coach, Belichick didn’t recite his résumé or make grandiose promises about winning national titles.
Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries; Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues; Convergent matrix — square ...