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Topic: Matrix inversion

Related Topics

Inverse matrix

Matrix (mathematics)

Normal matrix

LU factorization

Hermitian matrix

Orthogonal matrix

Symmetric matrix

Transpose

Matrix multiplication

Unitary matrix

Gauss algorithm

Determinant

Conjugate transpose

Hill cipher

Hermitian conjugate

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	Matrix inversion - Wikipedia, the free encyclopedia
		Matrix inversion is the following problem in linear algebra: given a square n-by-n matrix A, find a square n-by-n matrix B (if one exists) such that AB = BA = I
		An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert.
		Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
	en.wikipedia.org /wiki/Matrix_inversion (500 words)

	Invertible matrix - Wikipedia, the free encyclopedia
		A is row-equivalent to the n by n identity matrix I
		In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
		To check whether a given matrix is invertible, and to compute the inverse in small examples, one typically uses the Gauss-Jordan elimination algorithm.
	en.wikipedia.org /wiki/Invertible_matrix (485 words)

	Jacobian inversion
		One of the problems associated with the Jacobian inversion method is the matrix inversion.
		, the Jacobian matrix is rectangular and consequently not invertible.
		When the rank of the Jacobian matrix differs from the DOFs (some rows are linearly dependent), the number of solutions is infinite and the matrix is not invertible.
	www.cg.tuwien.ac.at /studentwork/CESCG/CESCG-2002/LBarinka/node16.html (323 words)

	Math 120 Calculator Instructions
		Enter the dimensions of the matrix (2 X 2) at the top if the dimensions are incorrect.
		Enter the dimensions of the matrix (2 X 1) at the top if the dimensions are incorrect.
		Matrix A is already selected as the default.
	lilt.ilstu.edu /cheri/math120/math120matrixinversion82or82.htm (246 words)

	Matrix Inversion - Uncyclopedia
		Matrix inversion consists of taking a matrix's O'Reilly Factor matrix and dividing by a number which is indeterminant.
		Successful matrix inversion can only be performed by beings with godlike powers, such as Oprah, Neo, or many calculators produced by Texas Instruments.
		Matrix Inversion often leads to the transformation of whole subspaces, or even acute attacks of deja vu.
	www.uncyclopedia.org /wiki/Matrix_Inversion (202 words)

	Matrix inversion
		Writing another special matrix of cofactorss, known as an adjoint matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
		where is the determinant of A, is the matrix cofactor, and represents the matrix transpose.
		In most practical applications, it is in fact not necessary to invert a matrix, but only to solve a system of linear equations.
	www.ukpedia.com /m/matrix-inversion.html (259 words)

	Matrix Operations
		The result is a new matrix with the same dimensions in which each element is the sum of the corresponding elements of the previous matrices.
		The inverse of a square matrix is a matrix of the same size that, when multiplied by the matrix, gives an identity matrix of the same size.
		Matrix inversion is often used to solve a set of simultaneous linear equations.
	www.stanford.edu /~wfsharpe/mia/mat/mia_mat2.htm (1904 words)

	[No title]
		is the determinant of the matrix obtained be omitting the first row and column j of A. This is a recursive definition.
		A determinant of 0 means that the matrix is singular and does not have an inverse.
		The characteristic Polynomial of a matrix A, is the determinant of xI - A for a variable x.
	www.people.vcu.edu /%7Edprimeau/cmsc621/fall02/presentations/tyson/tyson652.html (285 words)

	Matrix Inversion
		Multiplying by the original matrix gives you an identity matrix with small numerical errors.
		can usually not tell for certain whether or not the matrix is singular: all it can tell is for example that the determinant is small compared to the entries of the matrix.
		This takes the matrix, multiplies it by its inverse, and shows the first row of the result.
	documents.wolfram.com /v4/MainBook/3.7.6.html (296 words)

	Optical data processing systems and methods for matrix inversion, multiplication, and addition - Patent 4800519
		Alternatively, discrete matrix data may be impressed onto a data beam by spatially modulating the cross section of a data beam in terms of, for example, either its localized intensity or polarization vector.
		By treating the four quadrant field (5) as one matrix of order 2N, the terms of a new matrix may be calculated, using Gaussian elimination, by applying the formula: ##EQU5## where X.sub.nm.sup.new is the desired term of the new matrix, and X.sub.nm.sup.old is the corresponding term of the original matrix (5).
		The matrix A is stored in the upper left quadrant; the matrix B in the upper right quadrant; the matrix C (with element polarity inverted) in the lower left quadrant; and the matrix D in the lower right quadrant of the accumulator 16'.
	www.freepatentsonline.com /4800519.html (9351 words)

	System of Linear Equations and Matrix Inversion
		The coefficient matrix must be a squared-matrix appearing on the upper left corner of the Data-Matrix, therefore, do not leave any blank rows in between.
		To find the inverse of a square matrix of size n, solve n systems of equations with a unit vector as their right hand side.
		Notice that the coefficient of the variables X1 and X2 are matrix A in both systems of equations, however the RHS are two identity vectors in n=2 dimensional space.
	home.ubalt.edu /ntsbarsh/Business-stat/otherapplets/SysEq.htm (690 words)

	symmetric matrix inversion problems (Site not responding. Last check: 2007-11-03)
		I'm studying for a final in a wireless communications class, and one of the topics we're expected to be able to work is the calculation of the minimum mean squared error (MMSE) for an adaptive equalizer, given a certain input vector and equalizer filter structure.
		The formula we are supposed to use in calculating the MMSE thus involves inverting the 2x2 matrix R, and manipulating it along with other system parameters to arrive at a final result.
		I just concatenate an identity matrix of the same size as the matrix to be inverted with the original matrix; reduce it using elementary row operations so that the identity matrix stands on the left; and now the inverted matrix appears in the right half of the concatenated matrix.
	talkaboutelectronicequipment.com /group/comp.dsp/messages/160455.html (300 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-03)
		Date: 02/27/98 at 18:34:43 From: DuBois Ford Subject: Matrix Inversion by way of the Cayley-Hamilton Theorem I would like to know what the Cayley-Hamilton Theorem is and how it is used to find the inverse of a matrix.
		The characteristic equation is the equation whose roots are the eigenvalues of the matrix.
		If these terms are unfamiliar, I will illustrate with a 2x2 matrix, but the ideas can be generalized to an nxn matrix.
	mathforum.org /library/drmath/view/51978.html (244 words)

	Distributed Algorithms for Matrix Inversion for Heterogeneous Dynamic Networks (Site not responding. Last check: 2007-11-03)
		The size of the matrix is determined by the number of control points used to describe a part or set of parts to be considered.
		For all of these methods, each computer in the network is concerned with several rows of the matrix or, in the case very small computers, part of a row.
		The size of the matrix and the speed and type of network are all highly important in determining the best algorithm for a given situation.
	www.engin.umd.umich.edu /ceep/tech_day/1998/reports/CISreport1/CISreport1.html (979 words)

	Speeding up matrix inversions (Site not responding. Last check: 2007-11-03)
		The matrix P is derived from the SCFG rules and probabilities (either the left-corner relation or the unit-production relation).
		For an application using a fixed grammar the time taken by the precomputation of left-corner and unit-production matrices may not be crucial, since it occurs off-line.
		Even if the matrix P is sparse, the matrix inversion can be prohibitive for large numbers of nonterminals n.
	www.icsi.berkeley.edu /~stolcke/papers/cl95/node51.html (393 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		First, an $n\times n$ matrix $(A^{T} A)$ and a vector $(A^{T}~\vec{y})$ with length $n$ are initialised with zeroes.
		The system can be solved by matrix inversion as soon as at least $n$ equations have been accumulated.
		The accumulation matrix and vector are not affected, so accumulation may continue after a solve.
	aips2.nrao.edu /wyoung/tmp/lsfit.help (808 words)

	Matrix Inversion (Site not responding. Last check: 2007-11-03)
		There are a variety of published programs for doing matrix inversion, typically by elimination, described in the next section; for example, MS Basic came with example code for it, and of course, the Numerical Recipes books stuff.), although normally, the matrix isn't composed of complex numbers.
		The adjoint matrix is computed by taking the transpose of a matrix where each element is cofactor of the corresponding element in the source matrix.
		The cofactor is the determinant of the matrix created by taking the original matrix and removing the row and column for the element you are calculating the cofactor of.
	home.earthlink.net /~jimlux/radio/math/matinv.htm (1216 words)

	Matrix C++ template class library with full source code
		Matrix TCL Pro is a matrix template class library for performing matrix algebra calculations in C++ programs in easy and efficient manner for engineering / scientific works.
		It supports most of the matrix algebra operations and enables programmers to use matrix object just like other built-in data types in their C++ programs.
		It supports arithmetic operations, sub-matrix operations, inversion, various matrix decompositions, solution of simultaneous linear equations, eigen value and eigen vector problems, and much more.
	www.techsoftpl.com /matrix (189 words)

	Least-squares fit - matrix inversion (Site not responding. Last check: 2007-11-03)
		Matrix [A] now occupies the rectangular area with corners at cells (F3) and (N22) inclusive.
		This is usually due to the makeup of the matrix itself and not to a limitation of the software, although beginning with large UTM values that are then cubed and multiplied can frequently exceed the range, resulting in loss of precision and a matrix that can not be inverted.
		Prior to using this matrix inversion technique to obtain a polynomial "regional" gravity surface, it is necessary to convert standard map coordinates (usually latitude and longitude) into kilometers (usually east and north of some origin).
	www.eeescience.utoledo.edu /Faculty/Stierman/CM/Arch/4630/LABS/Invert.htm (1299 words)

	Matrix Inversion
		To be able to view the math in this document, use the PDF version, or please consider using another browser, such as Mozilla, Netscape 7 or above or Microsoft Internet Explorer 6 or above (MathPlayer required for IE).
		A matrix inverse has the property that when it is multiplied by the original matrix (on the left or on the right), the result will be the identity matrix.
		is the (n−1) x (n−1) matrix obtained from A by eliminating its i-th column and j-th row.
	cnx.rice.edu /content/m2113/latest (277 words)

	Lab for Matrix Inversion
		Construct the inverse for the matrix A, by using Gaussian elimination.
		The entries in the matrix for this exercise must be typed in by hand in order to make sure that only six decimal places are stored in the computer.
		Observe the difference between the computed inverse matrices when the Hilbert matrix is chopped.
	math.fullerton.edu /mathews/numerical/mi.htm (259 words)

	Matrix Inversion
		Mathematica TE can never tell for certain whether or not the matrix is singular; all it can tell is for example that the determinant is small compared to the entries of the matrix.
		Mathematica TE suspects that you are trying to invert a singular numerical matrix, it prints a warning.
		Mathematica TE prints a warning if you invert a numerical matrix that it suspects is singular.
	documents.wolfram.com /teachersedition/MathematicaBook/17.5.html (305 words)

	CS 130, Matrix Inversion (Site not responding. Last check: 2007-11-03)
		If the matrix has an inverse, the program will find the inverse and print the results of multiplying the original matrix by the inverse.
		The inverse of a matrix is another matrix that when you multiply those two matrices together you get the identity matrix as the result.
		then in the result matrix, row 1, column 0 is found by multiplying row 1 of matrix A by column 0 of matrix B and adding the terms together.
	students.cs.byu.edu /~cs130ta/labs/block_3/block3_2.html (355 words)

	TechiesAbode.com : Matrix Inversion
		Hi Hari Kishore, I am total zilch on matrix fundas.....so I am not sure if LU decomposition is the most efficient algorithm for matrix inversion and the likes.
		Matrix inversion using GJE is known to have about the same arithmetic cost as matrix inversion methods using Gaussian elimination (LU factorization) and have a connection with those methods.
		A quick test would be to use matlab and see if you can do matrix inversion on matrices of the order of 8000 on the same hardware on which your program is running.
	www.techiesabode.com /forum/posts_w.php?forum_id=119 (939 words)

	Routines for Matrix Inversion
		?getri computes the inverse of a general matrix that has been factored by ?getrf.
		?sytri computes the inverse of a symmetric matrix that has been factored by ?sytrf.
		?sptri uses packed storage for computing the inverse of a symmetric matrix that has been factored by ?sptrf.
	www.intel.com /software/products/mkl/docs/mklqref/matrixin.htm (236 words)

	Matrix Inversion!
		However, since A is a matrix, not a real number, we cannot prove it in the same way.
		The theorem is stating that if any row operations (switching rows, multiplying by a constant, or adding/subtracting a multiple of another row) taking some matrix to the identity exist, those very operations can bring the identity into the inverse of that matrix we started out with.
		There is an n × n matrix C such that CA = I. There is an n × n matrix D such that AD = I. is an invertible matrix.
	math.nyu.edu /~neylon/linalgfall04/project1/ewb/thms.html (475 words)

	SDL Delphi Component Suite - Invert (Matrix)
		The method Invert calculates the inverse of a quadratic matrix and stores the result on top of the original matrix (thereby destroying the original contents).
		The matrix inversion is carried out using the LU-decomposition procedure (decomposition into two triangle matrices).
		The lower limit of the necessary free heap memory can be calculated by the equation n((n+2)8+2) (bytes), with 'n' being the number of columns of the matrix.
	www.lohninger.com /helpcsuite/invert_(tmatrix).htm (127 words)

	Is Matrix inversion O(N^3)? (Site not responding. Last check: 2007-11-03)
		The original post was asking by the way, about computational cost of a matrix *inversion* algorithm, so I am not sure how matrix multiplication can help to clarify the original question.
		The whole point is that the algorithm is the same, whether the components of the matrix are scalars or block submatrices.
		Many algorithms (including some of the matrix inversion algorithms) take excactly a given number of operations as a function of the input data size.
	www.seriousliving.net /new-4473032-1134.html (10322 words)

	Matrix Inversion (Site not responding. Last check: 2007-11-03)
		adj(A) is formed by replacing each element of the matrix by its cofactor.
		The cofactor is the determinant of the matrix resulting from eliminating the row and the column of each element
		determinant is the number resulting from multiplying the terms of the matrix in a prescribed way.
	chemeng.uah.edu /courses/CHE244fa99/Class08/tsld010.htm (48 words)

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