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	Rank (linear algebra) - Wikipedia, the free encyclopedia
		The column rank of a matrix A is the maximal number of linearly independent columns of A.
		The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").
		It is equal to the linear rank of the derivative.
	en.wikipedia.org /wiki/Rank_(matrix_theory) (887 words)

	PlanetMath: rank of a linear mapping
		Composition of linear mappings does not increase rank.
		"rank of a linear mapping" is owned by yark.
		This is version 9 of rank of a linear mapping, born on 2002-02-19, modified 2006-01-13.
	planetmath.org /encyclopedia/RankLinearMapping.html (100 words)

	Matrix Reference Manual
		The dimensions of a matrix with 2 rows and 3 columns are specified as 2#3.
		Thus [1 2 3; 4 5 6] is a matrix with 2 rows and 3 columns.
		DIAG(a) is the diagonal matrix whose diagonal elements are the elements of a.
	www.psi.toronto.edu /matrix/intro.html (1063 words)

	PlanetMath: Sylvester's law
		The number of non-zero entries on the diagonal is known as the rank of the quadratic form.
		The rank and signature (whichever definition; pick your favorite) of a quadratic form are invariant under change of basis.
		This implies that two quadratic forms over the same finite-dimensional real vector space are related by a change of basis if and only if they have the same rank and signature.
	planetmath.org /encyclopedia/Rank5.html (258 words)

	Matrix laminates
		Matrix laminates are obtained by iterative lamination of an already built composite with the same initial materials at each step of the procedure.
		The scheme of assembling a matrix laminate structure: Material 1 is added to the obtained composite in differently oriented layers at each step.
		An anisotropic matrix laminates structure has the same effective properties as a laminate from the outer material and a matrix laminates made of remaining part of this material with the nucleus from the other one.
	www.math.utah.edu /~cherk/mathmat/workshop/theory/exact/node9.html (242 words)

	How Google Finds Your Needle in the Web's Haystack (Site not responding. Last check: 2007-11-03)
		What is needed is a means of ranking the importance of the pages that fit the search criteria so that the pages can be sorted with the most important pages at the top of the list.
		This has the effect of modifying the hyperlink matrix H by replacing the column of zeroes corresponding to a dangling node with a column in which each entry is 1/n.
		The matrix S has the pleasant property that the entries are nonnegative and the sum of the entries in each column is one.
	www.ams.org /featurecolumn/archive/pagerank.html (3061 words)

	15: Linear and multilinear algebra; matrix theory
		Classic topics in linear algebra and matrix theory are at the center of the diagram: 15A03: Vector spaces, 15A04: Linear transformations, 15A15: Determinants, and 15A21: Canonical forms (e.g.
		Furthest down is 15A72: Invariant theory and tensor algebra, which crosses to the study of invariants in Group Theory (20) and in polynomial rings (13: Commutative Algebra and 14: Algebraic Geometry).
		Eigenvalues of a symmetric matrix and the symmetric part of a general square matrix.
	www.math.niu.edu /~rusin/known-math/index/15-XX.html (1605 words)

	ISU Combinatorial Matrix Theory Research Group
		The nonzero-graph of A is the graph having as vertex set {1,...,n}, and, as its set of edges, the set of (unordered) pairs {i,j} such that both i and j are vertices with i not equal to j and aij not equal to 0.
		In recent years there has been a great deal of interest in the possible eigenvalues of a real symmetric matrix whose nonzero entries are described by a given graph or by the pattern of the signs of the entries, especially a tree (i.e., a connected graph with no cycles).
		A Fischer matrix is a P-matrix that satisfies Fischer's inequality for all principal submatrices.
	orion.math.iastate.edu /lhogben/research/mcgrg.html (1342 words)

	DMP: Definitions from Discrete Mathematics
		An adjacency matrix is a means of representing a graph in the form of a matrix.
		The Borda method is a group ranking method which assigns weights to individual preference schedules by giving a number of points for each first place vote another number for each second place vote and so on.
		The dimenstion of a square matrix is the number of rows (or columns in that matrix.
	www.colorado.edu /education/DMP/def.html (3248 words)

	INI Programme RMA Workshop - Special Week on Ranks of Elliptic Curves and Random Matrix Theory
		The connection between ranks of elliptic curves and random matrix theory arises from random matrix models for the values of L-functions in families.
		Random matrix theory gives an order of magnitude prediction for the number of vanishings mentioned above, but there is an unknown constant which is required to produce an asymptotic formula.
		It is hoped that random matrix theory can be used to predict the frequency with which curves in a family have a given rank.
	www.newton.cam.ac.uk /programmes/RMA/rmaw01.html (718 words)

	The Matrix Online for PC Review - PC The Matrix Online Review
		The Matrix is famous for its spectacular action sequences and its distinct sense of style.
		It's not that this is a game that only hardcore Matrix fans would enjoy (since they certainly might not), it's that it takes the same sort of person who can appreciate the entire movie trilogy in spite of its flaws to appreciate this game.
		Probably the most original trait of The Matrix Online is how it doesn't force players to lock themselves into a particular character profession, but instead allows them the freedom to mix and match different abilities at their whim.
	www.gamespot.com /pc/rpg/matrixonline/review.html (1489 words)

	Capital Area Theory Seminar (Site not responding. Last check: 2007-11-03)
		The Capital Area Theory Symposia is an UMIACS sponsored series of symposia in theoretical computer science bringing computer scientists from around the world to the Capital area.
		The rigidity of a matrix $A$ with respect to the rank bound $r$ is the minimum number of entries of $A$ that must be changed to reduce the rank of $A$ to or below $r$.
		The entries of these matrix families are (i) square roots of $n^2$ distinct primes and (ii) primitive roots of unity of prime orders for the first $n^2$ primes.
	www.cs.umd.edu /areas/Theory/CATS/catss06.html (1431 words)

	Rank - Wikipedia, the free encyclopedia
		Rank is a very broad term with several meanings.
		As an adjective it used to mean profuse, conspicuous, absolute, or unpleasant, especially in relation to the sense of smell or taste.
		Rank (formation) Military term for a line of soldiers.
	en.wikipedia.org /wiki/Rank (181 words)

	ESC - Risk Matrix (Site not responding. Last check: 2007-11-03)
		Risk Matrix is designed to allow the program/project team using this tool to continually update and reassess program risks based on the progress of risk mitigation plans.
		Risk Matrix is particularly good in the front-end; that is, in facilitating the identification of risk areas.
		Version 2.20 includes a major improvement involving the method by which risks are ranked (results are now more useful because they have less clustering and less ambiguity) and allows entry of more than 256 characters in a cell of text.
	akss.dau.mil /software/37.jsp (773 words)

	Rank (matrix theory)
		In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent.
		The column rank and the row rank are indeed equal and is simply called the rank of A.
		then the rank of A can also be defined as the dimension of the image of f, or as n minus the dimension of the kernel of f (see linear map for a discussion of image and kernel).
	www.fact-index.com /r/ra/rank__matrix_theory_.html (575 words)

	Evolution: "Theory" or "Fact"?
		In physics parlance, what makes it a theory is that it takes a small number of ideas (the gravity formula plus Newton's laws of mechanics) and explains in detail a lot of phenomena (orbiting of the planets around the sun, orbiting of the moons round the planets, various lab measurements, etc.).
		The theory of gravity has been engulfed by a theory of space and time so we now have a "mechanism" for gravity, at the expense of needing a "mechanism" for space and time, whatever that might mean.
		Actually a theory is a TOOL which scientists use for organizing and understanding their observations and which they also use for planning new experiments.
	www.mattox.com /genome/theory-fact.html (4226 words)

	ARCC Workshop: Spectra of families of matrices described by graphs, digraphs, and sign patterns
		The spectrum (i.e., the set of eigenvalues) of a matrix of data plays a vital role in many applications, and sometimes the entries of a data matrix are not known exactly.
		The spectrum of various matrices associated with a graph, such as the adjacency matrix and the Laplacian matrix, can also give information about the graph, or about an application that is modelled by a graph.
		The minimum rank of a tree is easily determined, but progress on other graphs has been slow.
	www.aimath.org /ARCC/workshops/matrixspectrum.html (514 words)

	Matrix2000.htm (Site not responding. Last check: 2007-11-03)
		MATRIX is a glaze storage system and formula calculation package with unique features not to be found in other glaze applications available at present.
		Matrix is now used internationally throughout the world by educational institutions, studio potters and industrial ceramists.
		A free copy of GlazeTeach is included with each purchase of Matrix V5 for PC GlazeTeach for both Windows and Mac is now available for those who wish to obtain their own copy of this popular glaze theory study course.
	www.matrix2000.co.nz (268 words)

	A Sequential Factorization Method for Recovering Shape and Motion from Image Streams
		A covariance-like matrix is stored instead of feature positions, and its size remains constant as the number of frames increases.
		First, the measurement matrix is factorized into two matrices of rank 3 using the singular value decomposition.
		The rank theory claims that the dimension of each subspace is at most three, and the first step of the factorization method finds those subspaces in the high dimensional input spaces.
	www-2.cs.cmu.edu /afs/cs/usr/tk/www/Projects_www/IUW94morita.fm.html (4030 words)

	Ecomatrix - PHP Web Hosting - PageRank algorithm
		The ranking algorithm is a function of on- and off-page factors.
		Wakfer's paper takes the position that Page Rank can't be bleed or lost and that all outbound links, whether internal or external to the domain, are given an equal PR vote.
		Bob Wakfer's paper states that the inbound links in his chart are voting 1/50th of their PR (have an average of 50 outbound links, internal or external), and are calculated as having a Page Rank of medium value.
	www.ecomatrix.eu /Articles/Internet/PageRank-algorithm.html (2022 words)

	in theory (Site not responding. Last check: 2007-11-03)
		The other is that, however cleaner our theories are when formulated asymptotically, we should not lose sight of the fact that the ultimate goals of complexity theory are finite results.
		If $G$ is Abelian (if the group operation is commutative), then the representation theory of $G$ is very simple and it is "just" the theory of Fourier transforms of functions $f: G \rightarrow C$ (where C are the complex numbers), which is much easier than it sounds.
		Marge, I agree with you - in theory.
	www.pkblogs.com /in-theory (6469 words)

	Project #1 Fishing in George's Bank
		Matrix algebra, notion of rank (chapter 4 esp. 4.4 of Kolman, chapter 3 Bretscher, chapter 3 Leon)
		The rank of the controllability matrix is easily shown to be three, so the system is controllable.
		B and determine whether the rank increases or not.
	www.prenhall.com /divisions/esm/app/ph-linear/kolman/html/proj8.html (1131 words)

	Rank Test Based Matrix Perturbation Theory
		Establishing the rank of a matrix is an important problem in a wide variety of econometric and statistical contexts that includes the classical identification problem in linear simultaneous equation models, determining the rank of demand systems, the order of lag polynomials in ARMA models, and the composition of factor models.
		We consider a rank test for a matrix for which a root-n consistent estimator is available whose limiting (normal) distribution's variance matrix may be of either unknown or less than full rank.
		Using Matrix Perturbation Theory, we determine that the asymptotic distribution of the test statistic is chi-squared.
	ideas.repec.org /p/sce/scecf0/153.html (350 words)

	Inertia Tensor
		In fact, a scalar is a tensor of zero rank:
		In addition, a vector is a tensor of the first rank since its transformation follows
		A 2-dimensional symmetric matrix is not necessarily a tensor of the 2nd rank.
	kwon3d.com /theory/moi/iten.html (237 words)

	LINEAR SYSTEMS THEORY - EE585
		The purpose of this course is to provide the students with the basic tools of modern linear systems theory: stability, controllability, observability, realization theory, state feedback, state estimation, separation theorem, etc. We develop the continuous and discrete-time cases in parallel.
		Elements of matrix algebra: determinant, rank, eigenvalues, matrix polynomials (3.6, p 62 of [1] and 2.1M, p 132 of [2]), Cayley-Hamilton Theorem (3.6, p 63, 65, 68 of [1])
		There is a typo in problem 3.17 of [1] (homework #4): the 1st row/2nd column of the matrix Q should be lT 2/2.
	www-scf.usc.edu /~hespanha/EE585 (1902 words)

	Matrix Theory: A Second Course
		They know Gauss-Jordan elimination and understand what a singular matrix is. They want a book that reviews the basics, but opens doors to more advanced topics that elementary books touch only lightly or not at all.
		It addresses the question of how to change a coordinate system in order to simplify a matrix—say to diagonalize the matrix, as is done to covariance matrices by principal component analysis.
		If you are looking for an advanced book that downplays matrices, and emphasized more abstract theory leading towards functional analysis, consider Lad (1996) instead.
	www.riskbook.com /titles/ortega_j_(1987).htm (270 words)

	[No title]
		See Michael James Wester, _Symbolic Calculation # and Expression Swell Analysis of Matrix Determinants and Eigenstuff_, Ph.D. # dissertation, University of New Mexico, Albuquerque, New Mexico, December # 1992, p.
		This matrix is notorious for causing # numerical eigenvalue routines to fail.
		261] Pi/2matrix([[2, 1, 1], [2, 3, 2], [1, 1, 2]]): map(Im, exponential(%I)); # Matrix square root => {+-[[3 1], [1 4]], +-1/sqrt(5) [[-1 7], [7 6]]} m:= matrix([[10, 7], [7, 17]]); evalm(sqrt(m)); # The following method is undocumented.
	math.unm.edu /~wester/demos/MatrixTheory/problems.maple (423 words)

	Read This: Google's PageRank and Beyond
		With this addition, the hyperlink matrix becomes stochastic, but this is not quite enough.
		Now we have a matrix — the Google matrix — that’s the sum of a sparse matrix and a rank one matrix; it is stochastic, irreducible and aperiodic.
		Google page rankings are said to be recomputed once a month in a process called the Google Dance.
	www.maa.org /reviews/GooglePageRank.html (1015 words)

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