Skew-symmetric matrix

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Topic: Skew-symmetric matrix

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	Matrix (mathematics) - Wikipedia, the free encyclopedia
		The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
		The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
		The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
	en.wikipedia.org /wiki/Matrix_(mathematics) (1577 words)

	linalg5.html
		For j>=i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument.
		For j>i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument.
		The lower triangle of the matrix is specified by the following relation: - M^* = transpose(M), were M^* denotes the matrix consisting of all the complex conjugates of M. This rules implies that the main diagonal is zero, therefore the argument should not include it.
	www.win.tue.nl /~amc/oz/om/cds/linalg5.html (1052 words)

	An introduction to MATRICES
		[7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6)
		A diagonal matrix is a square matrix with all de non-diagonal elements 0.
		An identity matrix I is a diagonal matrix with all diagonal element = 1.
	home.scarlet.be /~ping1339/matr.htm (921 words)

	G17
		is a constant matrix and the LHS is
		Our purpose of introducing matrix groups is to let the reader have a taste of Lie groups, which is an important mathematical subject.
		A general method for computing matrix exponential is given in Appendix A. Here we only consider simple examples for which this method is not needed.
	www.math.carleton.ca /~ckfong/G17/G17.html (750 words)

	PlanetMath: skew-symmetric matrix
		According to Jacobi's Theorem, the determinant of a skew-symmetric matrix of odd dimension is zero.
		This is version 3 of skew-symmetric matrix, born on 2001-11-21, modified 2003-05-08.
		Cross-references: odd, determinant, Jacobi's Theorem, imaginary, eigenvalues, vector space, operator, trace, transpose, properties, skew-Hermitian matrices, complex, skew-symmetric, matrix, real, dimension, square matrix
	planetmath.org /encyclopedia/SkewSymmetricMatrix.html (127 words)

	Matrix Definitions
		A square matrix is said to be symmetric if its elements are symmetric about the diagonal.
		The diagonal of the square matrix A is the top-left to bottom-right diagonal A
		A matrix with one column is called a column vector and a matrix with one row is called a row vector .
	www.maths.abdn.ac.uk /~igc/tch/engbook/node28.html (202 words)

	3.3 Matrix Precision and Shape
		A symmetric matrix is a square matrix with the property that the matrix is equal to its transpose.
		Symmetric matrices are often used to represent quadratic forms; for example, the energy in a system can be represented as a symmetric matrix.
		The complex analog of a symmetric matrix is a Hermitian matrix.
	www.roguewave.com /support/docs/hppdocs/lpkug/3-3.html (659 words)

	Matrix Manual: Special Matrices
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
		matrix, A, is an nn Toeplitz matrix* in which a(i,j) is a function of {(i-j) modulo n}.
		[22]: A 22 matrix is a Givens reflection iff it is a Householder matrix.
	www.ee.uwa.edu.au /~roberto/teach/matrix/special.html (2558 words)

	PlanetMath: Cayley's parameterization of orthogonal matrices
		These two formulae are each other's inverses and set up a one-to-one correspondence between orthogonal and skew-symmetric matrices.
		Hermitian operator or study symmetric extensions of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operator.
		The sole entry of a one dimensional unitary matrix must have modulus 1 and the sole entry of a one-dimensional Hermitean matrix must be real.
	planetmath.org /encyclopedia/CayleysParameterizationOfOrthogonalMatrices.html (572 words)

	Matrix Algebra
		Important properties of the transpose operation and triangular and symmetric matrices are contained in Theorem 1.5.3.
		The power of a square matrix, which is important in the subject of Markov chains (which we mentioned briefly in Sec.
		A matrix which has blocks separated by dotted or solid lines is called a partitioned matrix.
	distance-ed.math.tamu.edu /Math640/chapter1/node8.html (1646 words)

	ALMOST SKEW-SYMMETRIC MATRICES
		Using the notion of the numerical range, we obtain eigenvalue inequalities and a localization of the spectrum of an almost skew-symmetric matrix.
		We show that almost skew-symmetry is invariant under principal pivot transformation and inversion, and that the symmetric parts of Schur complements in almost skew-symmetric matrices have rank at most one.
		Almost skew-symmetric matrices are real matrices whose symmetric parts have rank one.
	math.la.asu.edu /~rmmc/rmj/vol34-1/MCD (80 words)

	Outcome Statement
		A. Define symmetric matrix, skew-symmetric matrix, and orthogonal
		Interpret an orthogonal matrix as a transformation that is a
		Prove that the determinant of an orthogonal matrix is 1 or -1.
	www.math.byu.edu /~math302/content/lessonoutl/lessonoutl47.html (115 words)

	Matrix Set PLATZ
		The original matrix is derived as the (negative) square of a purely imaginary skew-symmetric matrix.
		The larger matrix is a finite-difference model for the shallow wave equations for the Atlantic and Indian Oceans.
		The smaller matrix corresponds roughly to the North Atlantic Ocean.
	math.nist.gov /MatrixMarket/data/Harwell-Boeing/platz/platz.html (319 words)

	Pade.html
		This transformation is well known to be orthogonal by construction for skew-symmetric S. From a computational viewpoint, the block-matrix algorithm for the evaluation of C=C(S) is the result of manipulating the four matrix equations that are obtained from the blocked form of [I-S]*C=[I+S].
		The diagonal blocks of EXP(K) will be symmetric for even powers of K whereas odd powers of K will contribute to the skew-symmetric blocks.
		The matrix approximation to EXP(K) employed in SCF minimizations should be accurately orthogonal as well as appropriately close to the exponential value.
	www.emsl.pnl.gov /proj/tms/hpcc_actinides/research-98/argonne/Pade.html (479 words)

	Symmetric Matrices
		On rank estimation in symmetric matrices: the case of indefinite matrix estimato...
		On the concentration of eigenvalues of random symmetric matrices...
		On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalu...
	www.scienceoxygen.com /math/250.html (240 words)

	Lecture 9 --- 6.837 Fall '01
		Next, consider how the skew symmetric matrix of a vector generates a vector that is perpendicular to both the axis and it's input vector.
		From the idenity matrix in the third term it is easy to see how it will generate a vector in the same direction as the input vector.
	graphics.lcs.mit.edu /classes/6.837/F01/Lecture09/Slide16.html (51 words)

	Matrix Algebra
		The determinant of a square matrix is a single number calculated by combining all the elements of the matrix.
		An orthogonal matrix is a square matrix which produces a unit matrix if it is multiplied by its own transpose.
		A diagonal matrix is a square matrix in which all the elements are zero except for the elements on the leading diagonal, eg:
	www.morello.co.uk /matrixalgebra.htm (1255 words)

	Pfaffian
		The Pfaffian is an invariant polynomial of a skew-symmetric matrix (Note that it is not invariant under a general change of basis but rather under a proper orthogonal transformation).
		In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries.
		The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero.
	www.worldhistory.com /wiki/P/Pfaffian.htm (527 words)

	Determinant of the Sum of a Symmetric and a Skew-Symmetric Matrix
		The set of all possible determinant values of the sum of a (complex) symmetric matrix and a skew-symmetric matrix with prescribed singular values is determined.
		Determinant of the Sum of a Symmetric and a Skew-Symmetric Matrix : SIAM Journal on Matrix Analysis Vol.
		This set can also be viewed as the best containment region for the determinant of a square matrix $X$ in terms of the singular values of its symmetric and skew-symmetric parts.
	epubs.siam.org /sam-bin/dbq/article/29342 (141 words)

	algebra.txt
		Prove the fact that some combination of skew-symmetric matrix give a orthogonal matrix (sic).
		Express A in terms of S. Show that the eigenvalues of a skew matrix are imaginary.
		Standard Generals Questions - Algebra Matrices -------- Eigenvalues of a symmetric matrix.
	www.math.princeton.edu /graduate/generals/algebra.txt (924 words)

	Results
		A regular tournament matrix is a {0,1} square matrix with equal row sums.
		Definition: The adjacency matrix for a tournament graph is a tournament matrix.
		This gives us a biclique covering of 4 and so bc(T)=n-1Because the maximum number of independent ones, or the biclique covering number, which we know to be less than the boolean rank of the matrix is equal to n-1 the boolean rank of M must be greater than or equal to n-1.
	www-math.cudenver.edu /~rrosterm/project/node5.html (1283 words)

	Harbourne's Math 818 Problem set 3
		Note that A is a skew symmetric matrix.
		Prove or give an explicit counterexample: the nonzero roots of the characteristic polynomial of a real nxn skew symmetric matrix A are all purely imaginary.
		Let A be an m x m matrix with entries in a field F. Prove A is skew symmetric if and only if
	www.math.unl.edu /~bharbour/M818Spr05/probsets/ProbSet3.html (446 words)

	Denis Vanderstraeten (Abstract)
		The main idea of the preconditioner is to split the matrix in its symmetric and skew-symmetric part and to ``invert'' the (shifted) skew-symmetric matrix.
		The rates of convergence of iterative methods with standard preconditioning techniques such as ILUT usually degrade when the skew-symmetric component 'S' of the matrix is relatively large.
		In this paper, we address the issue of preconditioning matrices with such large skew-symmetric component.
	www-sccm.stanford.edu /~dv/pers/abstract/a5.html (159 words)

	Untitled Document
		] as the sum of symmetric and skew symmetric matrix
	www.classteacher.com /content/curriculum/boardpapers/classtwelve/maths98set1-1.shtml (96 words)

	Cholesky-like Factorizations of Skew-Symmetric Matrices - Benner, Byers, Fa, Mehrmann, Watkins (ResearchIndex)
		Abstract: Every real skew-symmetric matrix B admits Cholesky-like factorizations B = R T JR where J = h 0 I I 0 i.
		This paper presents a backward-stable O(n 3) process for computing such a decomposition, in which R is a permuted triangular matrix.
		...Hamiltonian matrix that is both J semide nite and non singular.
	citeseer.ist.psu.edu /374740.html (448 words)

	Notation of a Matrix and Operations with Matrices
		as a sum of a symmetric and a skew-symmetric matrix.
		can be expressed as a sum of a symmetric matrix and a skew-symmetric matrix.
		(i = 1 : m) of the transpose of matrix A ("pasting'' the matrices of the column-vectors or of the transposed row-vectors)
	www.math.ut.ee /~toomas_l/linalg/lin1/node12.html (172 words)

	shell.m
		function shell(a,k,xx) % Plots the shell (and spectrum) of an % almost skew-symmetric matrix `a' % `k' is the number of plotted points % `xx' is the character string passed on to spectrum plotting % Author: Panos Psarrakos and Michael Tsatsomeros if nargin
		0 aa=x+irt(1); bb=x+irt(2); plot(srreal(aa),imag(aa),'r.') hold on plot(srreal(bb),imag(bb),'r.') hold on end xlabel('Real Axis') ylabel('Imaginary Axis') end axis([-1 r+1 -n n]) spectrum(a,xx) else ' Matrix is not almost skew-symmetric' end
	www.sci.wsu.edu /math/faculty/tsat/files/matlab/shell.m (71 words)

	We consider the problem of the determination of a direction vector u which optimizes some utility function f(u,
		P is a skew symmetric matrix formed by the elements of u, called the dual to u matrix.
		Given a transformation matrix R we can determine the corresponding rotation axis u from P using Eq.
		In this case it follows that R has as diagonal matrix elements the following:
	www.mlahanas.de /Math/orientation.htm (530 words)

	631-1-98
		\\ (i) Show that if $n$ is odd, then $\det A=0$.\\ (ii) Find a $2\times 2$ skew-symmetric matrix $A$ with $\det A\neq 0$.
		Show that the columns of the matrix $X$ are the eigenvectors of the matrix $A$.
		(3 pts) (i) Show that if a square matrix $A$ is diagonalizable, then so is $A^2$.\\ (ii) Find a $2\times 2$ matrix $A$ such that $A^2$ is diagonalizable but $A$ is not.
	www.math.uab.edu /chernov/teaching/631-1-98 (371 words)

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Topic: Skew-symmetric matrix

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