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Topic: Hilbert matrix

	David Hilbert
		Hilbert studierte unter Ferdinand von Lindemann an der Albertina, wo er auch 1885 seine Dissertation 'Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunctionen' schrieb.
		Hilbert trug dazu bei, dass sein Freund Hermann Minkowski, den er in seiner Königsberger Zeit kennengelernt hatte, ebenfalls eine Professur in Göttingen erhielt.
		Hilberts Arbeiten zu Funktionenräumen (Hilbert-Raum) und partiellen Differentialgleichungen gehören heute zu den Grundlagen der mathematischen Physik.
	www.all2know.com /de/wikipedia/d/da/david_hilbert.html (0 words)

	Hilbert
		Characteristic of Hilbert's unabated, albeit justified, egotism, one of Hilbert's papers on invariants called Sylvester and Cayley "the representatives of the naive period" and himself the champion of "the critical period" of invariant theory.
		Hilbert stated that "the theory of number fields is an edifice of rare beauty and harmony." One important contribution from this report is his Satz 90, which is a theorem on relative cyclic fields.
		Hilbert defined a p-adic norm as an integer in the quadratic field K that is congruent to the norm of a suitable integer in K modulo any power of p.
	abyss.uoregon.edu /~js/glossary/hilbert.html (0 words)

	Matrix Inversion - Wolfram Mathematica
		When you invert an approximate numerical matrix, Mathematica can usually not tell for certain whether or not the matrix is singular: all it can tell is for example that the determinant is small compared to the entries of the matrix.
		This matrix is singular, but the warning is different, and the result is useless.
		This takes the matrix, multiplies it by its inverse, and shows the first row of the result.
	reference.wolfram.com /mathematica/tutorial/MatrixInversion.html (0 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		Every 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether its determinant is +1 or -1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (0 words)

	hilbert :: Functions (Signal Processing Toolbox)
		The Hilbert transformed series has the same amplitude and frequency content as the original real data and includes phase information that depends on the phase of the original data.
		The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the amplitude and frequency.
		The instantaneous amplitude is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle.
	www.mathworks.com /access/helpdesk/help/toolbox/signal/hilbert.html (0 words)

	Linear Algebra
		This command constructs a diagonal matrix, that is a square matrix whose off-diagonal entries are all zero.
		The minor is the determinant of the matrix obtained from M by deleting the i-th row and the j-th column.
		The Wronskian matrix is created by putting each function as the first element of each column, and filling in the rest of each column by the (i-1)-th derivative, where i is the current row.
	homepage.mac.com /yacas/manual/refchapter9.html (0 words)

	Essays/Hilbert Matrix - J Wiki
		The Hilbert matrix is a square matrix whose (
		the determinant of the Hilbert is the reciprocal of an integer
		The permanent of the inverse Hilbert matrix is
	www.jsoftware.com /jwiki/Essays/Hilbert_Matrix (381 words)

	Hilbert matrix - The Jiggies Reference Guide (Site not responding. Last check: )
		In linear algebra, a Hilbert Matrix is a matrix with elements H
		The Hilbert matrix is derived from the integral
		The Hilbert matrices are the canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation.
	www.jiggies.com /reference/Hilbert_matrix (132 words)

	Linear Algebra
		This command constructs a diagonal matrix, that is a square matrix whose off-diagonal entries are all zero.
		The minor is the determinant of the matrix obtained from M by deleting the i-th row and the j-th column.
		The Hilbert matrix is extremely sensitive to manipulate and invert numerically.
	yacas.sourceforge.net /refchapter9.html (2078 words)

	LinBox: TemplateLinBox::Hilbert< _Field, Trait > class Reference (Site not responding. Last check: )
		This is a class of n by n Hilbert matrices templatized by the field in which the elements reside.
		The third is chosen be defualt to be the {LinBox} vector trait of the vector.
		Because the Hilbert matrix is symmetric, this is the same as calling the apply function.
	www.eecis.udel.edu /~linbox/doc/html/classLinBox_1_1Hilbert.html (294 words)

	Matrix Manual: Special Matrices
		matrix, A, is an nn Toeplitz matrix* in which a(i,j) is a function of {(i-j) modulo n}.
		A is symmetric iff it is congruent to a diagonal matrix.
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.psi.toronto.edu /matrix/special.html (0 words)

	Linear algebra
		Note: GSchmidt returns a matrix with orthogonal columns that span the same subspace as the columns of mat, provided the columns of mat are linearly independent.
		A Hilbert matrix is a special case of the Hankel matrix.
		The determinant of an orthogonal matrix is ±1.
	www.ibiblio.org /technicalc/packages/mathtools/linearalgebra.htm (1176 words)

	Matrix Generator HILBERT
		This is a Java applet which generates the N-by-N Hilbert matrix with elements A(i,j) = 1/(i+j-1), a famous example of a badly conditioned matrix.
		This generator was developed by Bruce Miller of NIST based on Nick Higham's Test Matrix Toolbox for MATLAB.
		The Matrix Market is a service of the Mathematical and Computational Sciences Division / Information Technology Laboratory / National Institute of Standards and Technology.
	math.nist.gov /MatrixMarket/data/MMDELI/hilbert/hilbert.html (0 words)

	PlanetMath: Hilbert matrix
		An example of an inverted Hilbert matrix when
		For more fun with Hilbert matrices, see [1].
		This is version 3 of Hilbert matrix, born on 2002-09-28, modified 2005-06-24.
	planetmath.org /encyclopedia/HilbertMatrix.html (0 words)

	MathViews - Functions
		Hence the balanced matrix is inv(T)exprT. When two return arguments are specified, then T is the diagonal transformation matrix, and B is the balanced matrix inv(T)exprT. This function is meaningful for complex arguments as well as real arguments, and it returns a real value in either case.
		The upper triangular matrix is the transpose of
		The argument must be a square matrix; if not you must use the svd function to find the matrix's singular values instead.
	www.mathwizards.com /techinfo/man/mathv05.htm (4971 words)

	PyMatrix.matrix
		Build a matrix from a sequence of arrays or matrices.
		Return a Householder matrix - an orthgononal, unitary and hermitian (and thus symmetric) matrix.
		other: a matrix, a list of matrices, or objects which can be converted into matrices.
	www3.sympatico.ca /cjw/PyMatrix/Doc/matrix-summary.html (0 words)

	PlanetMath: Frobenius matrix norm
		See Also: matrix norm, matrix p-norm, vector norm, vector p-norm, Schur's inequality, trace, transpose, transpose, matrix logarithm
		Cross-references: size, unitary matrix, square matrix, property, conjugate transpose, definition, square, columns, rows, function, norm, matrices, valuation, ring
		This is version 15 of Frobenius matrix norm, born on 2001-10-06, modified 2007-06-24.
	planetmath.org /encyclopedia/HilbertSchmidtNorm.html (0 words)

	Research Technologies at Indiana University
		The Hilbert matrix is a favorite example of numeric (as opposed to symbolic) linear algebra.
		The Hilbert matrix is badly conditioned, which means that it corresponds to a system of linear equations which are hard to solve accurately.
		A curious feature of the Hilbert matrix is that the entries of its inverse are all integers.
	www.indiana.edu /~statmath/math/maple/linalg.html (0 words)

	Matlab Primer (1)
		The elements within a row of a matrix may be separated by commas as well as a blank.
		If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations (for addition, subtraction, and division as well as for multiplication) in which case each entry of the matrix is operated on by the scalar.
		Evaluation of the expression produces a matrix, which is then displayed on the screen and assigned to the variable for future use.
	math.ucsd.edu /~driver/21d-s99/matlab-primer1.html (0 words)

	Linear algebra
		Notes: Cholesky returns a lower-triangular matrix instead of an upper-triangular one, because trying to transpose the lower-triangular matrix results in conj() being applied to all symbolic elements, which looks ugly.
		Note: GSchmidt returns a matrix with orthogonal columns that span the same subspace as the columns of mat, provided the columns of mat are linearly independent.
		A Hilbert matrix is a special case of the Hankel matrix.
	www.technicalc.org /packages/mathtools/linearalgebra.htm (0 words)

	GNU Octave - Matrix Manipulation
		For a matrix argument, return a row vector of ones and zeros with each element indicating whether any of the elements of the corresponding column of the matrix are nonzero.
		function does not allow sort keys to be specified, it can't be used to order the rows of a matrix according to the values of the elements in various columns(6) in a single call.
		Return a matrix with random elements uniformly distributed on the interval (0, 1).
	sunsite.univie.ac.at /textbooks/octave/octave_16.html (0 words)

	3×3 Matrix Explorer - Wolfram Demonstration
		The trace of a matrix is the sum of the entries on the main diagonal (upper-left to lower-right).
		The determinant is computed from all the entries of the matrix and is nonzero precisely when the matrix is nonsingular, that is, when the equation
		The matrix rank is the number of linearly independent columns and is equal to three precisely when the matrix is nonsingular.
	demonstrations.wolfram.com /33MatrixExplorer (0 words)

	Linear Algebra on TI-8x Graphing Calculators
		Consider the matrix equation Ax=b, where A is the 2x2 matrix* [[1,2][3,4]] and b is the vector [1,1].
		We examine the classic ill-conditioned matrix, the Hilbert matrix, whose (i,j) entry is 1/(i+j).
		A matrix equation involving the Hilbert matrix can be difficult to solve - the ill-conditioned nature of the matrix can lead to large errors during reduction.
	www.math.umbc.edu /~campbell/Calculators/TI8x/linalg.html (0 words)

	test matrices
		We saw the second difference matrix when we were approximating the second derivative on equally spaced data.
		The Hilbert matrix is related to the interpolation problem on the interval [0,1].
		The Hilbert matrix is at once nice because its inverse has integer elements and also not nice because it is extremely difficult to compute the inverse using the usual formulæ for matrix inverses.
	www.metu.edu.tr /~smerdan/testmatrices.htm (0 words)

	Lab for Matrix Inversion
		Construct the inverse for the matrix A, by using Gaussian elimination.
		The entries in the matrix for this exercise must be typed in by hand in order to make sure that only six decimal places are stored in the computer.
		Observe the difference between the computed inverse matrices when the Hilbert matrix is chopped.
	mathews.ecs.fullerton.edu /numerical/mi.htm (0 words)

	The Matrix Inverse
		This is not a error, there is a problem with the matrix, it is ill-conditioned.
		The Hilbert matrix is a classic example of an
		Calculations involving the Hilbert matrix and its inverse sometimes cannot be trusted.
	math.fullerton.edu /mathews/n2003/inversematrix/InverseMatrixMod/Links/InverseMatrixMod_lnk_5.html (0 words)

	Matrix Inversion
		Multiplying by the original matrix gives you an identity matrix with small numerical errors.
		can usually not tell for certain whether or not the matrix is singular: all it can tell is for example that the determinant is small compared to the entries of the matrix.
		suspects that you are trying to invert a singular numerical matrix, it prints a warning.
	documents.wolfram.com /v4/MainBook/3.7.6.html (0 words)

	QNT \| Product - Demo applications
		Here are small examples of using ExLAF77: solution of system of equations with Hilbert matrix and finding eigenvalues of Van der Monde matrix.
		This makes inverting Hilbert matrix a challenge even at small n, if native-precision arithmetics is used.
		As a reference point, for the Hilbert matrix of n=100 you may wish to specify a mantissa of 30 words.
	www.quasinewtonian.com /product_demo.html (0 words)

	Determinants, Part 4
		A Hilbert matrix is an n x n matrix H whose (i,j,)th entry is 1/(i+j-1).
		Enter the definition in the worksheet to see a typical Hilbert matrix.
		Enter the commands in your worksheet to plot the column vectors of the 3 x 3 Hilbert matrix.
	www.math.duke.edu /education/ccp/materials/linalg/determinants/dete4.html (0 words)

	Hilbert Matrices
		We are primarily interested in Hilbert matrices because they are very badly conditioned, even for small values of n, and because their condition number grows rapidly with n.
		The ill-conditioning nature of the Hilbert matrices can be traced back to the approximation problem which we used to introduce them.
		This means that the rows of the Hilbert matrix are very linearly dependent, i.e., that the matrix is very nearly singular.
	www.cs.unc.edu /~dm/UNC/COMP205/LECTURES/LINALG/lec10/node1.html (0 words)

	Math 343 - Numerical Methods - MatLab Assignment 5
		Matrix H is called the Hilbert matrix which is a famous example of a badly conditioned matrix.
		To generate a Hilbert matrix of order N, the command is hilb(N).
		From the results, you may see that as N is getting larger the Hilbert matrix is getting closer to be singular very quickly.
	www.mathcs.citadel.edu /~chenm/234.dir/mlsav.lis (0 words)

	Math 696 -- Introduction to the linear algebra package
		Still another way to define a matrix is to specify a formula for the entries in terms of the row and column indices.
		Since A is a 2×3 matrix, it is not possible to multiply A by itself.
		These commands compute (a) the inverse of a three-digit numerical approximation to the 4×4 Hilbert matrix and (b) the exact inverse of the 4×4 Hilbert matrix.
	www.math.tamu.edu /~harold.boas/courses/math696/Maple-linear-algebra.html (0 words)

	Homework 1 Assignment 1 -- Fall 1998Au=b
		This assignment deals with a linear system of equations, Ax=b, where A is the Hilbert matrix.
		Consider a system of n linear equations of the form Ax = b where the matrix A is the Hilbert matrix of order n.
		The entries in the hilbert matrix are defined to be:
	cauchy.math.colostate.edu /Homework/Hw_1/hw_1.html (0 words)

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