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Topic: Gramian matrix

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	ED231A: Matrix Arithematic 3 (Site not responding. Last check: 2007-10-13)
		The rank of a pxq matrix is the largest number r such that there exists a set of r rows or columns which is linearly independent.
		All the eigenvalues of nonsingular Gramian matrices are positive.
		Any symmetric matrix whose principal minor determinants are all non-negative is a Gramian matrix (This is the Existence Theorem).
	www.gseis.ucla.edu /courses/ed231a1/notes4/mat32.html (433 words)

	ED231A: Matrix Arithematic 3 (Site not responding. Last check: 2007-10-13)
		The matrix of v's is used to define the transformation to maximum variance.
		of the eigenvlaues of a matrix A is equal to the trace of A, tr(A).
		of the eigenvlaues of a matrix A is equal to the determinant of A, Det(A).
	www.gseis.ucla.edu /courses/ed231a1/notes4/mat31.html (857 words)

	Hilbert matrix - Wikipedia, the free encyclopedia
		In linear algebra, a Hilbert matrix is a matrix with unit fraction elements
		Hilbert also mentions the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer which he expresses as the discriminant of a certain hypergeometric polynomial related to the Legendre polynomial.
		The Hilbert matrix is also totally positive (meaning the determinant of every submatrix is positive).
	en.wikipedia.org /wiki/Hilbert_matrix (379 words)

	[No title]
		The next step is to perform a Galerkin projection onto the states corresponding to the largest singular values of the balanced gramians for the region of interest in state-space [2].
		The output argument xhat represents the observability gramian (flag = 0) or covariance matrix (flag ~= 0) This routine applies initial condition perturbations in each state to compute the empirical observability gramian or the observability covariance matrix.
		The input argument yhat represents the (empirical) controllability gramian or covariance matrix, xhat refers to the (empirical) observablility gramian or covariance matrix, and n is the number of states.
	cheweb.tamu.edu /orgs/groups/Hahn/Model_Reduction/documentation.doc (2629 words)

	Overlap matrix - Wikipedia, the free encyclopedia
		The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system.
		In addition, if the basis vectors form an orthonormal set, the overlap matrix will be the identity matrix.
		The overlap matrix is always n×n, where n is the number of basis functions used.
	en.wikipedia.org /wiki/Overlap_matrix (122 words)

	Math Forum Discussions (Site not responding. Last check: 2007-10-13)
		An nxn real matrix A is said to be a Gramian matrix if
		there exists a real (square) matrix B such that
		A=(B^t)B. Prove that A is a Gramian matrix iff A is
	mathforum.org /kb/thread.jspa?messageID=4139384&tstart=0 (89 words)

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