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Topic: Nonsingular matrix

	Glossary of Linear Algebra Terms
		The column space of a matrix is the subspace spanned by the columns of the matrix considered as vectors.
		The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of c.
		The row space of a matrix is the subspace spanned by the rows of the matrix considered as vectors.
	www.caam.rice.edu /~zhang/caam310/glossary.html (1005 words)

	PlanetMath: similar matrix
		is similar to the diagonal matrix of its eigenvalues
		, with the matrix of its eigenvectors acting as the similarity transformation.
		This is version 11 of similar matrix, born on 2002-02-20, modified 2006-11-30.
	planetmath.org /encyclopedia/SimilarMatrix.html (263 words)

	Inverse of structured rank matrices
		Suppose we have a matrix for which the (first) subdiagonal is different from zero, and all the other entries in the strictly lower triangular part are zero.
		matrix plus a block diagonal matrix for which the sizes of the blocks of the first and the latter diagonal are the same.
		The inverse of a lower bidiagonal matrix is a lower triangular semiseparable matrix and vice versa.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node9.html (1090 words)

	Math 310 - Glossary of Linear Algebra Terms
		The column space of a matrix is the subspace spanned by the columns of the matrix considered as a set of vectors.
		An elementary matrix is a matrix that is obtained by performing an elementary row operation on an identity matrix.
		The row space of a matrix is the subspace spanned by the rows of the matrix considered as a set of vectors.
	www.math.uic.edu /math310/glossary.html (1060 words)

	The Inverse of a Matrix
		An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
		An elementary matrix is a matrix that is obtained by performing a single elementary row operation to an identity matrix.
		An elementary row operation is the process of either (1) replacing one row of a matrix with the sum of itself and a multiple of another row (2) Interchanging two rows (3) Multiplying all entries in a row by a nonzero constant.
	www.math.nyu.edu /~neylon/linalgfall04/project1/jja/group7.htm (1427 words)

	Polynomial Matrix Glossary
		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
		The roots or zeros of a polynomial matrix P are those points in the complex plane where P loses rank.
	www.polyx.com /glossary.htm (1119 words)

	[No title]
		Matrix ill-conditioning is a general term used to describe a rectangular matrix of values which is unsuitable for use in a particular analysis.
		In this type of Matrix Plot, 2D Scatterplots are arranged in a matrix format (values of the column variable are used as X coordinates, values of the row variable represent the Y coordinates).
		The column (or row) rank of a rectangular matrix of values (e.g., a sums of squares and cross-products matrix) is equal to the number of linearly independent columns (or rows) of elements in the matrix.
	www.statsoft.com /textbook/glosm.html (5420 words)

	ED231A: Matrix Arithematic 3
		The matrix of v's is used to define the transformation to maximum variance.
		of the eigenvlaues of a matrix A is equal to the trace of A, tr(A).
		of the eigenvlaues of a matrix A is equal to the determinant of A, Det(A).
	www.gseis.ucla.edu /courses/ed231a1/notes4/mat31.html (857 words)

	Matrices and solutions of systems of simultaneous equations
		The determinant of the matrix A is denoted
		bc, is the determinant of the matrix at the left of (*).
		The matrix in the numerator of each expression is obtained by replacing the column in the matrix on the left of () that corresponds to the variable for which we are solving with the column vector on the right of ().
	www.chass.utoronto.ca /~osborne/MathTutorial/MAT.HTM (393 words)

	INV Function (Site not responding. Last check: 2007-10-09)
		The INV function produces a matrix that is the inverse of matrix, which must be square and nonsingular.
		All matrix elements less than or equal to sing are now considered rounding errors of the largest matrix elements, so they are taken to be zero.
		For example, if a diagonal or triangular coefficient matrix has a diagonal value less than or equal to sing, the matrix is considered singular by the DET, inv, and SOLVE functions.
	www.asu.edu /sas/sasdoc/sashtml/iml/chap17/sect125.htm (267 words)

	Generalized permutation matrix - Wikipedia, the free encyclopedia
		In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e.
		A nonsingular matrix A is a generalized permutation matrix if and only if it can be written as a product of a nonsingular diagonal matrix D and a permutation matrix P:
		The set of n×n generalized permutation matrices with entries in a field F forms a subgroup of the general linear group GL(n,F), in which the group of nonsingular diagonal matrices Δ(n, F) forms a normal subgroup.
	en.wikipedia.org /wiki/Monomial_matrix (269 words)

	Matrix as Linear Transformation (Site not responding. Last check: 2007-10-09)
		The rank of a matrix a is the dimension of the subspace spanned by the vectors in a.
		The identity matrix is square and nonsingular, and its rows span every possible vector of length n.
		The matrix spance the entire vector space, and is a basis for that space.
	www.mathreference.com /la,lxmat.html (549 words)

	PA = LU Factorization with Pivoting (Site not responding. Last check: 2007-10-09)
		A sufficient condition for the factorization to exist is that all principal minors of A are nonsingular.
		We have seen in Example 3 an example of a nonsingular matrix A could not be directly factored as A = LU.
		An n×n permutation matrix P is a matrix with precisely one entry whose value is "1" in each column and row, and all of whose other entries are "0." The rows of P are a permutation of the rows of the identity matrix and P can be written as
	math.fullerton.edu /mathews/n2003/LUFactorMod.html (469 words)

	Solving matrix equations using matrix division (Site not responding. Last check: 2007-10-09)
		, which creates a matrix with entries from a uniform distribution on the interval (0,1).
		When used with a nonsquare matrix, the backslash operator solves the appropriate system in the least-squares sense; see
		Of course, as with the other arithmetic operators, the matrices must be compatible in size.
	www.math.mtu.edu /~msgocken/intro/node8.html (162 words)

	Approximate Matrix Inverses and the Condition Number
		It states that a nonsingular matrix has widely different approximate right and left inverses if, and only if, it is ill-conditioned.
		The matrix norm corresponding to a vector norm is the real-valued function A
		A matrix with a high condition number is said to be ill-conditioned.
	efgh.com /math/invcond.htm (1135 words)

	[No title]
		Two important properties of convex sets to be utilized in constructing a solution scheme are as follows: 1) Each linear equation of a linear model forms a convex set 2) The intersection of two convex sets is also a convex (and thus the set of simultaneous linear equations will form a convex set.
		Nonsingular matrix: A nonsingular matrix is a matrix whose determinant does not equal zero.
		For an inverse to exist, the square matrix must be nonsingular.
	www.srl.gatech.edu /education/ME6103/LP-terms.doc (234 words)

	Semi-inverses
		is, in a sense, a matrix “halfway between” a nonsingular matrix A and its inverse.
		As a special case, for a one-by-one matrix A = (a) regarded as isomorphic to a scalar, for a > 0 we may observe that since matrix inversion corresponds in this case to reciprocation, the result of applying the operator Φ may be called the (principal) “semi-reciprocal” of the number a.
		Also, since the semi-inverse of any matrix in U can be polynomially generated, and since it is well known that when λ is an eigenvalue of a matrix A and p(x) is any polynomial it follows that p(λ) is an eigenvalue of p(A), we have:
	www.blackmesapress.com /Semi-inverses.htm (1328 words)

	The exponential matrix
		, the identity matrix, is the exponential matrix
		Not every matrix A has a basis of eigenvectors so not every matrix can be diagonalized.
		A Jordan block is a square matrix having the form
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node6.html (218 words)

	4.1 Matrix Decompositions
		One of the most important matrix transformations is the QR decomposition.
		Consequently, SVD can be used to find an effective rank of a matrix, to check a near singularity and to compute the condition number of a matrix.
		In previous sections, we described how matrix decompositions can be used for solving systems of linear equations.
	www.quantlet.com /mdstat/scripts/csa/html/node36.html (1953 words)

	The nullity theorem
		The theorem can be seen as an extension of the standard formula for calculating the inverse of a matrix, for which each element is determined by a minor in the original matrix.
		This lemma already implies the nullity theorem for square subblocks and for nullities equal to 1, since this case is equivalent with the vanishing of a determinant.
		In the next section we will use the previously obtained results about the ranks of complementary blocks of a matrix and its inverse to prove the rank properties of the inverse for some classes of structured rank matrices.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node8.html (389 words)

	Inverse of a Square Matrix
		In the first case, it is said to be singular and in the second case nonsingular.
		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.
	mat.gsia.cmu.edu /QUANT/notes/node12.html (365 words)

	VII. REVIEW OF LINEAR ALGEBRA*
		A square matrix A has at most one inverse; that is, the inverse of a matrix is unique, if it exists.
		For any mxm matrix A there exists a nonsingular matrix B such that BA is in echelon form.
		Hessian determinant is the determinant of the matrix of second and cross partials.
	courses.temple.edu /economics/notes/algebra/NBUCK1.htm (1359 words)

	The Order of a Matrix (Site not responding. Last check: 2007-10-09)
		If x is a large matrix, it is often worth while looking for a nonsingular matrix q that diagonalizes x.
		Therefore x is similar to a matrix y in jordan canonical form.
		Write the block as the sum of the diagonal matrix and the subdiagonal matrix.
	www.mathreference.com /la-mpoly,order.html (525 words)

	PSD Matrices and LLS
		to indicate that a matrix is PSD (and a similar notation for PD matrices).
		from the factorization as the square-root of the matrix
		In summary we find that solving the linear least squares problem involves solution of a system of equations involving a PD matrix.
	www.ee.ucla.edu /~brien/Rec7_PSDMatricesAndLLS.htm (1039 words)

	Matrix Operations
		The exception to this rule is the addition (subtraction) of a scalar to (from) a matrix.
		In this case the scalar is added to or subtracted from each element of the matrix individually.
		If is a square nonsingular matrix and b is a vector with as many rows, MATLAB evaluates the expression
	www.cs.ubc.ca /spider/cavers/MatlabGuide/node7.html (338 words)

	[No title]
		A Toeplitz matrix is defined by EMBED Equation.3 , which means that all entries on the same downward diagonal are required to be the same.
		Here is an example of a Hankel matrix: EMBED Equation.3 A Vandermonde matrix begins with a column vector, and raises the entries of the column to the c-1 power, where c is the number of the column in the matrix.
		MNEG: Input: Matrix M Output: Additive inverse of M The sum of M and the additive inverse of M is 0.
	www.cs.caltech.edu /~joyjoy/CSReport.doc (2247 words)

	Special Matrix Types
		It is almost always true that many of the zero entries in a matrix needn't be stored.
		A matrix with a small bandwidth will thus require little storage, enabling one to consider bigger problems on any finite-memory machine.
		Furthermore, many matrices with small bandwidths have, in addition, other nice properties, for example, strong diagonal dominance, etc. Researchers spend much time optimizing codes that are storage efficient and also have nice mathematical properties which enable them to produce more efficient solvers.
	www.physics.arizona.edu /~restrepo/475A/Notes/sourcea/node68.html (314 words)

	[No title]
		How can the inverse of a matrix be used to write down the solution of a system of linear equations?
		What is the determinant of a matrix when any two of its rows or columns are equal?
		What is the rank of a nonsingular matrix?ª )ó8 ¨Positive Definite Matricesª ¨¿When is a matrix positive definite?
	www.stanford.edu /~lambers/cs137/Lecture2.ppt (897 words)

	Application of Determinant to Systems: Cramer's Rule
		We have seen that determinant may be useful in finding the inverse of a nonsingular matrix.
		We can use these findings in solving linear systems for which the matrix coefficient is nonsingular (or invertible).
		where A is the matrix coefficient, B the nonhomogeneous term, and X the unknown column-matrix.
	sosmath.com /matrix/determ2/determ2.html (259 words)

	Some Theorems and Finding the Inverse of a Matrix (Site not responding. Last check: 2007-10-09)
		Theorem: A matrix is nonsingular if and only if its GJSF is the identity matrix.
		Technically, this is only a left inverse, since we only have multiplication on the left to get I. However, we KNOW A is nonsingular, so Elem must be the "real" inverse as well (this is not as clear if it is not known in advance that A is nonsingular).
		Theorem: Two matrices A and B are row equivalent if and only if B=EA for some nonsingular matrix E. Proof: If A and B are row equivalent, then there is a finite sequence of elementary row operations leading from A to B. As we saw above, this led to the matrix E (Elem), producing B=EA.
	www.ma.iup.edu /projects/CalcDEMma/linalg1/linalg104.html (527 words)

	[No title]
		The rank of a matrix is not changed if one row (column) is multiplied by a nonzero constant or if such a multiple of one row (column) is added to another row(column).
		A square matrix of order n is nonsingular iff it is of full rank rank(A) = n or, equivalently, (A ((0 The rank of a matrix is unchanged by premultiplying or postmultiplying by a nonsingular matrix.
		Thus if there are nonsingular matrices E and F for which EMBED Equation.3 Inverse If A is a square, nonsingular matrix of order n, a unique inverse matrix EMBED Equation.3 of order n exists, where EMBED Equation.3 .
	www.personal.kent.edu /~mqi/econ1/EconI2.doc (557 words)

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