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	Matrix (mathematics) - Wikipedia, the free encyclopedia
		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 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 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)

	PlanetMath: matrix
		As a rule, though, the word matrix and the notation associated with it are used only in connection with linear mappings.
		to be equipotent.) Square matrices naturally arise in connection with a linear mapping of a space into itself (called an endomorphism), and in the related case of a change of basis (from one basis of some space, to another basis of the same space).
		One more illustration: The derivative (at a point) of a mapping from one surface to another is a linear mapping; it is not a matrix of partial derivatives, because the matrix depends on a choice of basis but the derivative does not.
	planetmath.org /encyclopedia/Matrix.html (557 words)

	Trace (linear algebra) - Wikipedia, the free encyclopedia
		In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.
		A matrix whose trace is zero is said to be traceless or tracefree.
		The concept of trace of a matrix is generalised to the trace class of bounded linear operators on Hilbert spaces.
	en.wikipedia.org /wiki/Trace_(matrix) (611 words)

	Matrix Algebra
		Elements (entries) of the matrix are referred to by the name of the matrix in lower case with a given row and column (again, row comes first).
		(The main or principal diagonal in matrix B is composed of elements all equal to 1.) With a square, symmetric matrix, the transpose of the matrix is the original matrix.
		Multiplying a matrix by the identity matrix is analogous to the real number operation of multiplying a number or variable by 1: the resulting output is identical to the numbers input.
	luna.cas.usf.edu /~mbrannic/files/regression/matalg.html (1775 words)

	PlanetMath: square matrix
		A square matrix has the same number of rows as columns.
		is a three-by-three matrix with elements that are complex numbers.
		This is version 3 of square matrix, born on 2002-01-13, modified 2003-12-11.
	planetmath.org /encyclopedia/SquareMatrix.html (76 words)

	6.3 - The Inverse of a Square Matrix
		The inverse of a square matrix A, denoted by A
		Row-reduce (I suggest using pivoting) the matrix until the left side is the Identity matrix.
		You may multiply or divide a matrix by a scalar (real number) and the determinant is a scalar.
	www.richland.edu /james/lecture/m116/matrices/inverses.html (1467 words)

	6.4 - The Determinant of a Square Matrix
		The determinant of a square matrix A is denoted by "det A" or
		The matrix of minors is the square matrix where each element is the minor for the number in that position.
		The inverse of a matrix is found by dividing the adjoint of the matrix by the determinant of the matrix.
	www.richland.edu /james/lecture/m116/matrices/determinant.html (2597 words)

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

	[No title]
		A matrix has usually at least 2 x 2 dimensions, a vector has 1 x p or n x 1 dimensions, and a scalar has 1 x 1 dimensions and is therefore a single number.
		Any matrix that is multiplied by an appropriately dimensioned unit vector (where all elements are 1's) can be reduced in one dimension, and the new elements are sums of the previous ones.
		Multiplying a data matrix A by an appropriate diagonal matrix D "rescales" the A matrix.
	darkwing.uoregon.edu /~bfmalle/613/L3.html (1057 words)

	Magic Squares 2
		matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or either main diagonal is equal to the magic constant defined as:
		Addition of magic squares does not result in another magic square, so they cannot be added.
		A magic square is composite if it is the product of two strictly smaller magic squares.
	home.att.net /~srschmitt/magicsquares2.html (273 words)

	Punarworn.com - Mathematics web site
		The sum of the diagonal elements of a square matrix A is called the trace of A.
		The matrix of order n x m obtained by interchanging the rows and columns of an m x n matrix A is called the transpose of A and is denoted by A¢ (A transpose).
		If a matrix A is reduced to B by the use of elementary row transformations a lone, B is said to be row equivalent to A and conversely.
	www.punarworn.com /view.asp?ccode=2508 (570 words)

	[No title] (Site not responding. Last check: 2007-08-19)
		Example of a square matrix: B = EMBED Equation.3 Vector A special case of a matrix is a vector (or a column vector).
		Identity Matrix, Denoted I, is a diagonal matrix with 1's on the diagonal.
		Rank of a Matrix The rank of a matrix A, r(A), is the maximum number of linearly independent columns (vectors) or rows (transposes of vectors) of the matrix.
	statweb.calpoly.edu /jdaly/stat324/matrices.doc (1885 words)

	Inverse of a Square Matrix (Site not responding. Last check: 2007-08-19)
		The cofactor matrix is the matrix of determinants of the minors A
		The i,j'th minor of A is the matrix A without the i'th column or the j'th row.
		It can also be verified that the original matrix A multipled by its inverse gives the identity matrix (all zeros except along the diagonal which are ones).
	astronomy.swin.edu.au /~pbourke/analysis/inverse (149 words)

	Square Matrix (Site not responding. Last check: 2007-08-19)
		In a square matrix the number of rows equals the number of columns.
		A row matrix consists of a single row.
		Some books call a column matrix a column vector and call a row matrix a row vector.
	chortle.ccsu.ctstateu.edu /VectorLessons/vmch13/vmch13_4.html (133 words)

	Introduction to Econometrics - Matrix Algebra
		A special diagonal matrix is a matrix where all diagonal elements are equal to one (identity matrix denoted I).
		If A is a non singular square matrix then the roots of the inverse of A are equal to the reciprocal values of the roots of A but the eigenvectors are the same.
		The determinant of an orthogonal matrix is equal to 1 or -1.
	www.resacorp.com /matrix_algebra.htm (1824 words)

	Introduction and Basic Operations
		The size of the matrix, as a block, is defined by the number of Rows and the number of Columns.
		In this case, we say that the matrix is a (mxn) matrix (pronounce m-by-n matrix).
		Indeed, if we consider the matrix with all its entries equal to 0, then it is easy to check that this matrix has behavior similar to the number zero.
	www.sosmath.com /matrix/matrix0/matrix0.html (658 words)

	Math Forum - Ask Dr. Math
		Date: 03/24/2003 at 10:47:27 From: John Subject: Magic squares Let M be an integer-valued 3x3 matrix whose entries form a magic square.
		Let s be the sum of all entries in M and d be the determinant of M. Show that d/s is an integer.
		If you remember that in a magic square, a + e + i = g + e + c, you should be able to complete the proof.
	mathforum.org /library/drmath/view/62569.html (242 words)

	Inverse of a Square Matrix
		Interestingly, linear independence of vectors plays a role here: a matrix is singular if its columns form a set of linearly dependent vectors; and it is nonsingular if its columns are linearly independent.
		When A is a square nonsingular matrix, this linear system has a unique solution, which can be obtained as follows.
		and we are done, since the matrix in front of B is the identity.
	mat.gsia.cmu.edu /QUANT/notes/node12.html (365 words)

	Determinant of Square Matrix
		The determinant of a 2 by 2 matrix.
		The determinant of a 3 by 3 matrix.
		Cross out the row and column the first number is in.
	www.mecca.org /~halfacre/MATH/determinant.htm (272 words)

	Determinant of an arbitrary square matrix (Site not responding. Last check: 2007-08-19)
		of the matrix A is the n-1 by n-1 matrix made by the rows and columns of A except the i'th row and the j'th column is not included.
		If the determinant is 0 the matrix said to be "singular".
		A singular matrix either has izero elements in an entire row or column, or else a row (or column) is a linear combination of other rows (or columns).
	astronomy.swin.edu.au /~pbourke/analysis/determinant (232 words)

	MAPLE: Matrix Algebra
		To obtain the characteristic polynomial of a square matrix C:
		As long as the identity matrix appears in the coefficient matrix portion, then the rightmost column gives the solution x, y, z.
		n coefficient matrix and b is the n-term constant vector by:
	www.gmi.edu /acad/scimath/appmath/maple/linalg6.html (315 words)

	Determinant and Inverse of Matrices
		Finding the inverse of a matrix is very important in many areas of science.
		is defined as the determinant of the square matrix of order (n-1) obtained from A by removing the row number i and the column number j multiplied by
		Define the adjoint of A, denoted adj(A), to be the transpose of the matrix whose ij
	www.sosmath.com /matrix/inverse/inverse.html (190 words)

	AoPS Math Forum :: View topic - Square and matrix(not the moovie)
		Prove that there exists a {0,1} nxn matrix (which means an nxn matrix with 0 or 1 elements) whose square is the matrix all of whose elements are 1 if and only if n is a perfect square.
		You divide the nxn square in n smaller squares of dimension kxk.
		Then, consider the jxn matrix B formed only by the lines 1 to j.
	www.artofproblemsolving.com /Forum/post-5632.html (558 words)

	Computing Square Roots Of A Matrix - Kim (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		11 1.2.2 Existence of Square Roots of a Singular Matrix.
		0.3: Approximating The Logarithm Of A Matrix To Specified..
		6 Stable iterations for the matrix square root (context) - Higham - 1997
	citeseer.ist.psu.edu /kim97computing.html (518 words)

	[No title]
		A. Bjorck and S. Hammarling, A Schur method for the square root of a matrix, Linear Algebra and Appl., 52/53 (1983), pp.
		For general theory concerning square root of a positive (semi)definite transformation on a Hilbert space, see the book "Functional Analysis" by F.
		For algorithms and references for numerical methods, see the biblical "Matrix Computations" (3rd edition) by G. Golub and C. Van Loan, in particular section 4.2.10 and chapter 11, which is about matrix functions in general.
	www.math.niu.edu /~rusin/known-math/00_incoming/sqrt_mat (715 words)

	Error 1305. This operation is possible only for a square Matrix. (Site not responding. Last check: 2007-08-19)
		This operation is possible only for a square Matrix.
		The current matrix is not square, and you have tried to perform an operation that is only possible for a square matrix.
		The following methods require a square matrix: Cholesky, Determinant, Eigen, EigenC, Inverse, LU, Solve, SolveGauss, Trace
	www.bluebit.gr /matrix/version_3/Error1305.htm (64 words)

	GrandMaster Square >>> HomePage
		Whether you want to buy or sell antiquarian or just old chess stuff, this place is for you.
		GM Square >>> Questions for GM Alexander Baburin.
		You'll receive e-mail notification of updates on the site, info on new products, special offers, etc. We do not anticipate these mailing to be more frequent than once or twice a week.
	www.gmsquare.com (271 words)

	square matrix (Site not responding. Last check: 2007-08-19)
		Definition: A n × n matrix, i.e., one whose size is the same in both dimensions.
		Go to the Dictionary of Algorithms and Data Structures home page.
		Paul E. Black, "square matrix", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/squarematrix.html (68 words)

	Square Matrix Determinant (Site not responding. Last check: 2007-08-19)
		After you click the following "compute" button, you can input the elements of Matrix A
		The numerical results will be displayed in another new window!
		First input the coefficients of matrix A as:
	www.hpcsoft.com /products/MathSoL/matrix/matrixDeterminant.html (42 words)

	Inverse of the Square Wave Matrix (ResearchIndex)
		Abstract: The N \Theta N matrix A, whose ijth element A i;j, 1 i; j N, is defined by A i;j = (\Gamma1) dj=ie+1 ; is called the "square wave matrix." The ith row of A represents a \Sigma1- valued "square wave" function of j with half-period i.
		This paper shows that the inverse of the square wave matrix is sparse and its elements may be expressed in terms of the Mobius function from number theory; it follows that multiplying an N-vector by either A \Gamma1 or A (thus converting a waveform into a...
		@misc{ grabiner-inverse, author = "David Grabiner", title = "Inverse of the Square Wave Matrix", url = "citeseer.ist.psu.edu/42905.html" }
	citeseer.ist.psu.edu /42905.html (259 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #4680129 - LEAST SQUARE MATRIX METHOD FOR ANALYZING NEUTRON SPECTRA
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		For a journal article, please see the Resource Relation field.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4680129 (105 words)

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