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	Unimodular matrix
		In mathematics, a unimodular matrix is a square matrix with determinant +1 or -1.
		A totally unimodular matrix is a matrix for which every square non-singular submatrix is also unimodular.
		An integer program whose constraint matrix is totally unimodular can be solved efficiently since its LP relaxation gives rise to integer solutions.
	www.brainyencyclopedia.com /encyclopedia/u/un/unimodular_matrix.html (243 words)

	Unimodular matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a unimodular matrix is a square (A rectangular array of elements (or entries) set out by rows and columns) matrix with (A determining or causal element or factor) determinant +1 or -1.
		An (Click link for more info and facts about integer program) integer program whose constraint matrix is totally unimodular and whose right hand side is integral, can be solved efficiently since its (A long-playing phonograph record; designed to be played at 33.3 rpm) LP relaxation gives rise to integer solutions.
		Those properties are (Click link for more info and facts about sufficient) sufficient for a matrix to be totally unimodular (but they are not necessary).
	www.absoluteastronomy.com /encyclopedia/u/un/unimodular_matrix.htm (250 words)

	Unimodular matrix - Wikipedia, the free encyclopedia
		In mathematics, a unimodular matrix is a square matrix with determinant +1 or −1.
		A totally unimodular matrix is an integer matrix for which every square non-singular submatrix is also unimodular.
		An integer program whose constraint matrix is totally unimodular and whose right hand side is integral, can be solved efficiently since its LP relaxation gives rise to integer solutions.
	en.wikipedia.org /wiki/Unimodular_matrix (213 words)

	DESCRIPTION OF CALC (Site not responding. Last check: 2007-10-08)
		This performs a modification of LLL which is essentially a limiting form of sgcd(N) for large N. It is superior to egcd() in that it avoids inputting a large initial unimodular matrix and instead builds one from the identity matrix at the outset.
		The unimodular matrix P of a recent paper is sent to lllgcdmat.out.
		The output unimodular matrix P of paper is sent to lllhermitetrans.out.
	www.itee.uq.edu.au /~gap/calc/krm_calc.html (2347 words)

	Other `rcop' operations (Site not responding. Last check: 2007-10-08)
		An operation performed by `rcop' is the same as multiplication on the left or right by a matrix obtained by changing one 0 in an identity matrix to a nonzero number.
		For example, adding 3 times the first column to the last in a matrix with 5 columns is the same as multiplying on the right by the matrix, call it J, obtained by changing the entry in the upper right corner of the 5x5 identity matrix from 0 to 3.
		Note that when the identity matrix is reached, it is the upper left 5x5 corner of a matrix G (you can look at all of G simply by entering the command G at the Matlab prompt) that also containes matrices M and N for which MJN is the identity matrix.
	www.nyu.edu /classes/edwards/chap2/otherrcop.html (410 words)

	LLLGCD algorithm: gcd of 5 integers (Site not responding. Last check: 2007-10-08)
		gcd(2,5,14,23,29) alpha = 1 The unimodular matrix found is 0 1 -2 1 0 2 -1 -1 1 0 -1 2 -1 -1 1 2 -1 -2 0 1 1 1 0 1 -1 Call the rows b[1],..,b[5].
		Two other randomly found examples: gcd(103,500,1005,204,60) The unimodular matrix is 3 0 -1 4 -2 4 -2 0 2 3 4 1 0 -3 -5 -2 -5 2 4 -2 1 -3 2 -3 0 b[5]=[1,-3,2,-3,0] is a multiplier of lengthsquared 23.
		The unimodular matrix is -1 0 -1 2 1 0 3 -1 -2 0 2 2 -3 2 -2 1 4 4 3 -3 2 -3 -2 -1 -1 b[5]=[2,-3,-2,-1,-1] is a multiplier of lengthsquared 19.
	www.numbertheory.org /lll/example5.html (243 words)

	Reduction of Matrices and Lattices
		Given a matrix X belonging to the matrix module S=Hom_R(M, N) or the matrix algebra S=M_n(R), where R is a subring of the real field, compute a matrix Y whose non-zero rows are a LLL-reduced basis for the Z-lattice spanned by the rows of X (which need not be Z-linearly independent).
		By default the rows of the returned matrix Y are sorted so that all the zero rows are at the bottom and the non-zero rows are sorted first with respect to their norms (so the row with the smallest norm is at the top) and then with respect to lexicographical order.
		Given a matrix X belonging to the the matrix module S=Hom_R(M, N) or the matrix algebra S=M_n(R), where R is Z or Q, compute a matrix Y whose rows form a Seysen-reduced basis for the Z-lattice spanned by the rows of X. The rows of X need not be Z-linearly independent.
	www.math.lsu.edu /magma/text810.htm (2959 words)

	Mathematical Programming Glossary Page 2 (Site not responding. Last check: 2007-10-08)
		This arises in (linear) integer programming because it implies a basic solution to the LP relaxation is integer-valued (given integer-valued right-hand sides), thus obtaining a solution simply by a simplex method.
		An example of a totally unimodular matrix is the node-arc incidence matrix of a network, so basic solutions of network flows are integer-valued (given integer-valued supplies and demands).
		A nonsingular matrix whose Hermitian adjoint equals its inverse (same as orthogonal for real-valued matrices).
	carbon.cudenver.edu /~hgreenbe/glossary/second.php?page=U.html (491 words)

	Polynomial Matrix Glossary
		A polynomial matrix P has full column rank (or full normal column rank) if it has full column rank everywhere in the complex plane except at a finite number of points.
		The column leading coefficient matrix of P is the constant matrix whose (i, j) entry is the coefficient of the term with power
		The row leading coefficient matrix of P is the constant matrix whose (i, j) entry is the coefficient of the term with power
	www.polyx.com /glossary.htm (1119 words)

	Canonical Forms
		The elementary divisors of the matrix a belonging to a submodule of the module M_n(S).
		The (generalized) Jordan canonical form for the matrix a belonging to the algebra M_n(K), where the coefficient ring K must be a field allowing factorization of univariate polynomials over it.
		The rational canonical form of a matrix a belonging to M_n(K), where the coefficient ring K must be a field allowing factorization of univariate polynomials over it.
	www.math.lsu.edu /magma/text849.htm (1001 words)

	TJS paper
		Section 3 discusses unimodular loop transformations and introduces loop stretching; in addition, we argue that unimodularity is not sufficient for optimizing communication with tiling for distributed memory machines.
		A distance matrix is a collection of dependence distance vectors for a nested loop -- the distance vectors are the columns of the distance matrix.
		Barnett and Lengauer [5] conclude that unimodularity is not essential; to derive code for non-unimodular transformations, they generate piecewise loop bounds which are not exact.
	www.ee.lsu.edu /jxr/revised/revised.html (5041 words)

	[No title]
		A feedback (matrix) for this system is any m by n R-matrix K. The closed-loop system obtained applying feedback K to the system (F, G) is by definition the new system (F + GK, G).
		Let R be a Bezoul ring such that whenever a matrix A has content R, there is a vector v with A v unimodular.
		Let v be unimodular in kerA Then v is in F- 1G, and by Corollary 3.6, R is a PA-ring.
	www.math.rutgers.edu /pub/sontag/bssv.txt (4670 words)

	Canonical Forms for Elements
		A unimodular matrix T such that T.X = H, i.e., T is the product of elementary matrices which transforms X into Hermite normal form.
		If R is the ring of integers Z and the matrix T is requested (i.e., if an assignment statement is used with two variables on the left side), then the LLL algorithm will be used by default to improve T (using the kernel of X) so that the size of its entries are very small.
		Unimodular matrices P and Q such that P * X * Q = S, i.e., P and Q are matrices which transform X into Smith normal form.
	www.math.wisc.edu /help/magma/text521.html (609 words)

	Linear Algebra Glossary
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	orion.math.iastate.edu /burkardt/papers/linear_glossary.html (13535 words)

	Matrix Transformation is Complete for the Average Case
		Matrix Transformation is Complete for the Average Case:SIAM Journal on Computing Vol.
		In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time.
		The problem is this: Given a unimodular matrix $X$ of integers, a set $S$ of linear transformations of such unimodular matrices and a natural number $n$, decide if there is a product of $\leq n$ (not necessarily different) members of $S$ that takes $X$ to the identity matrix.
	epubs.siam.org /sam-bin/dbq/article/23207 (179 words)

	Distributional Matrix Correspondence
		The distributional matrix correspondence problem was first shown to be complete for DistNP by Gurevich [Gur90].
		We assume that all unimodular matrices of the same size have an equal chance to be selected.
		matrix is greater than 1, the minor component of the major column cannot be zero.
	www.uncg.edu /mat/acc-forum/avgnp/node32.html (775 words)

	Simplicial signed decompositions
		Therefore once we have a unimodular cone decomposition, the rational generating function of the original cone is a signed sum of ``simplicial'' rational functions.
		The first matrix has negative determinant which is not the same sign as the determinant of its parent matrix
		of cones in the decomposition is bounded polynomially in terms of the determinant of the input matrix.
	www.math.ucdavis.edu /~latte/theory/lattE/node3.html (1296 words)

	Abstract (Site not responding. Last check: 2007-10-08)
		By converting these components into a matrix polynomial form we show that the invertibility conditions can be described in terms of matrix rational approximants for a matrix power series determined from the entries of the mosaic matrix.
		In special cases these matrix rational approximations are closely related to Pad\'e and various well-known matrix-type Pade' approximants.
		We also show that the inversion components can be described in terms of unimodular matrix polynomials.
	www.cs.ualberta.ca /~cabay/Abstracts/inverse_mosaic.html (175 words)

	[No title]
		Basicly, I'm looking for a description > > of the n-step process for running a unimodular transformation > > of an input source code, given a transformation > > matrix T. Unimodular transformations are handled by a library, not a pass.
		Skweel is just one particular pass that calls the library to do the unimodular transformations it has chosen.
		Then it would call the transform library and pass it your matrix and a pointer to the SUIF code for that loop nest.
	suif.stanford.edu /suif/mlists/suif-talk/199407/19940723.html (758 words)

	Publications of Thanos Antoulas
		It explores the close relationships between a number of topics like: matrix linear fractional representations, nested feedback interconnections formal power series, recursive realization of matrix impulse response sequences matrix continued fractions a matrix Euclidean algorithm, etc. As a corollary the celebrated Berlekamp-Massey algorithm is extended to the matrix case.
		The natural tool for achieving this is a certain polynomial unimodular matrix of size p+m which can be associated to every system with m inputs and p outputs.
		The construction of solutions of the matrix rational interpolation problem in state-space form, is derived using the (block) Loewner matrix.
	www.ece.rice.edu /~aca/Pub.html (3353 words)

	[ref] 25 Integral matrices and lattices
		These operations compute the Smith normal form of a matrix with integer entries, using the strategy specified in the name.
		Also in the record are components holding the sign of the determinant, signdet, and the Rank of the matrix, rank.
		is either a square symmetric matrix or lower triangular matrix (only the entries in the lower triangular half are used by the program).
	www.math.niu.edu /help/math/gap4/ref/CHAP025.htm (1503 words)

	COSC5319:FINAL EXAM. (Site not responding. Last check: 2007-10-08)
		B. Show the inverse of a unimodular matrix is also unimodular.
		C. Show that if a row (J) of a unimodular matrix is replaced by the sum of itself and K (Integer) times another row (L), where L
		Find all solutions to: 3X + 2Y - 2Z = 10 in terms of a unimodular matrix U such that U*T represents all solutions where T = (10, t2, t3) and Ti is an independent variable which may be a variable of the equation.
	hal.lamar.edu /~KOH/5319/5319FINAL.HTML (278 words)

	Distributional Matrix Correspondence Problem (Site not responding. Last check: 2007-10-08)
		For convenience, when we mention unimodular matrix in this paper we mean that its dimension is two-by-two.
		In the definition of the distributional matrix correspondence problem, if we only allow positive unimodular matrices, where by ``positive" we mean that every entry is a positive integer, then we obtain the DISTRIBUTIONAL POSITIVE MATRIX CORRESPONDENCE PROBLEM.
		The distributional matrix correspondence problem was shown to be average-case NP-complete by Gurevich [Gur90] under a randomized reduction (see also [BG95]).
	www.uncg.edu /mat/acc-forum/avgnp/node17.html (142 words)

	Inversion of polynomial matrices
		When A is unimodular, the inverse is again a polynomial matrix; otherwise the inverse resides in
		Repeating the steps for the remaining triangular block we eventually produce the identity matrix.
		Therefore, performing the same opeartions simultaneously on the identity matrix, we finally make it into desired inverse.
	web.quick.cz /Robert.Hornych.2/algoritmos/Inversion.htm (232 words)

	INFORMS Miami 2001 Annual Meeting Session SC09 (Site not responding. Last check: 2007-10-08)
		A supernormal configuration is a Hilbert-basis H where any cone generated by a subset of H has its Hilbert basis in H. This provides a generalization of unimodular configurations.
		In particular, we prove that the chamber complex of H is equal to the Groebner fan of associated integer programs.
		For a unimodular matrix A and an integral vector b, we consider the counting function phi_A(b)={x : Ax=b x>=0 }.
	www.informs.org /Conf/Miami2001/TALKS/SC09.html (244 words)

	A Characterisation of the Matroids Representable (Site not responding. Last check: 2007-10-08)
		It follows from a fundamental (1958) result of Tutte that a binary matroid is representable over the rationals if and only if it can be represented by a unimodular matrix, that is, by a matrix over the rationals with the property that all subdeterminants belong to {0,1,-1}.
		For an arbitrary field F it is of interest to ask for a matrix characterisation of those matroids representable over F and the rationals.
		It is shown that a ternary matroid is representable over the rationals if and only if it can be represented over the rationals by a matrix A with the property that all subdeterminants of A belong to the set {0, 2^i: i an integer}.
	www.mcs.vuw.ac.nz /maths/papers/GPW1.html (178 words)

	UW - Computer Sciences Events
		Gabber and Galil showed how to construct expanders from certain unimodular (determinant 1) 2x2 matrices with integer entries.
		It is shown that every parabolic or hyperbolic fractional linear transformation explicitly defines an expander of bounded degree and constant expansion.
		Thus all but a vanishingly small fraction of unimodular matrices define expanders.
	www.cs.wisc.edu /abstracts/9-14-0_0.html (169 words)

	ACM TOMS: Bibliographic record for `Geurts:1997:AFP' (Site not responding. Last check: 2007-10-08)
		Every polynomial matrix is unimodularly equivalent to a column-reduced polynomial matrix.
		In this article a subroutine is described that takes a polynomial matrix $P$ as input and yields on output a unimodular matrix $U$ and a column-reduced matrix $R$ such that $PU = R$; actually, $PU - R$ is near zero.
		The performance of the subroutine on every example tried is satisfactory in the sense that the magnitude of the elements of the residual matrix $PU-R$ is about $\parallel P \parallel \parallel U \parallel EPS$, where $EPS$ is the machine precision.
	math.nist.gov /toms/cgi-bin/TOMSbibget?Geurts:1997:AFP (249 words)

	Chapter 2. Equivalence of Matrices (Site not responding. Last check: 2007-10-08)
		The main algorithm finds, for a given matrix A, unimodular matrices M and N for which MAN is a diagonal matrix.
		When you are done transforming the augmented matrix, you can use the Matlab function unhat to break it apart and find the diagonal matrix D, along with the unimodular matrices M and N.
		The function algo automatically applies the algorithm of Section 7 to a matrix in hat format.
	www.nyu.edu /classes/edwards/chap2.html (212 words)

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