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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 ...
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 ...
Positive semidefinite matrix; Positive semidefinite quadratic form; Positive semidefinite bilinear form This page was last edited on 2 May 2021, at 17:28 (UTC). Text ...
The Gram matrix of R is defined to be the matrix G = RR T. From this definition it follows that G. is an integer matrix, is symmetric, is positive-semidefinite,
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
An matrix is said to be positive semidefinite if it is the Gram matrix of some vectors (i.e. if there exist vectors , …, such that , = for all ,). If this is the case, we denote this as M ⪰ 0 {\displaystyle M\succeq 0} .
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.
Equality in this bound is attained for a real matrix N if and only if N is a Hadamard matrix. A positive-semidefinite matrix P can be written as N * N, where N * denotes the conjugate transpose of N (see Decomposition of a semidefinite matrix). Then