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Topic: Diagonal matrix

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Invertible matrix

Symmetric matrix

List of linear algebra topics

Sparse graph

Unitary matrix

Matrix decomposition

Pseudoinverse

Singular value decomposition

Adjacency matrix

Conjugate transpose

Spectral theorem

Eigenvalue

Differential operator

Pontryagin duality

Eigenfunction

	Diagonal matrix - Wikipedia, the free encyclopedia
		Any diagonal matrix is also a symmetric matrix, a triangular matrix, and (if the entries come from the field R or C) also a normal matrix.
		A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I.
		Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or linear map by a diagonal matrix.
	en.wikipedia.org /wiki/Diagonal_matrix (455 words)

	Tridiagonal matrix - Wikipedia, the free encyclopedia
		In linear algebra, a tridiagonal matrix is one that is "almost" diagonal.
		To be exact, a tridiagonal matrix has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.
		Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties.
	en.wikipedia.org /wiki/Tri-diagonal_matrix (219 words)

	PlanetMath: diagonal matrix (Site not responding. Last check: 2007-11-07)
		The identity matrix and zero matrix are diagonal matrices.
		A square matrix is diagonal if and only if it is triangular and normal (see this page).
		This is version 6 of diagonal matrix, born on 2003-06-28, modified 2004-03-07.
	planetmath.org /encyclopedia/DiagonalMatrix.html (151 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)

	Matrix Reference Manual
		The dimensions of a matrix with 2 rows and 3 columns are specified as 2#3.
		Thus [1 2 3; 4 5 6] is a matrix with 2 rows and 3 columns.
		DIAG(a) is the diagonal matrix whose diagonal elements are the elements of a.
	www.psi.toronto.edu /matrix/intro.html (1063 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)

	Special Matrices: Triangular, Symmetric, Diagonal
		We have seen that a matrix is a block of entries or two dimensional data.
		A symmetric matrix is a matrix equal to its transpose.
		A diagonal matrix is a symmetric matrix with all of its entries equal to zero except may be the ones on the diagonal.
	www.sosmath.com /matrix/matrix3/matrix3.html (345 words)

	PlanetMath: characteristic matrix of diagonal element cross-section (Site not responding. Last check: 2007-11-07)
		A diagonal element of an upper triangular matrix is of course an eigenvalue of that matrix, so the particular
		"characteristic matrix of diagonal element cross-section" is owned by lars_h.
		This is version 1 of characteristic matrix of diagonal element cross-section, born on 2005-09-02.
	planetmath.org /encyclopedia/CharacteristicMatrixOfDiagonalElementCrossSection.html (198 words)

	Possible Results of Similarity Transforms
		Ideally we try to diagonalize the matrix A by similarity transform: B = 1/TAT,since then the diagonal elements are the eigenvalues of A and we are done.
		The diagonal elements are the eigenvalues and the eigenvectors are the unit coordinate vectors e[k] = (0 0...
		The diagonal elements of A are the eigenvalues.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/50.htm (1424 words)

	Maybe this Explains the Economic Cycle... best Diagonal Matrix (Site not responding. Last check: 2007-11-07)
		A special diagonal matrix is a matrix where all diagonal elements are equal to one (identity matrix denoted I).
		Diagonal Matrix -- from MathWorld Diagonal Matrix -- from MathWorld A diagonal matrix is a square matrix \mathsf{A} of the form a_{ij} = c_i\delta_{ij}, where \delta_{ij} is the Kronecker delta, c_i are constants, and i,j=1, 2,..., n, with...
		An identity matrix is a special diagonal matrix in which all of the diagonal elements are equal...
	ascot.pl /th/Fourier3/Diagonal-Matrix.htm (507 words)

	Creating a Matrix (Site not responding. Last check: 2007-11-07)
		A matrix is a rectangular array of numbers (or functions!) arranged in horizontal rows and vertical columns.
		The entries of a matrix A are written aij, where i denotes the row and j the column.
		An identity matrix is the diagonal matrix with diagonal consisting of all 1's.
	www.ma.iup.edu /projects/CalcDEMma/linalg0/linalg001.html (229 words)

	Matrix Manual: Special Matrices
		A symmetric or Hermitian Hessenberg matrix is tridiagonal.
		A is symmetric iff it is congruent to a diagonal matrix.
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.ee.uwa.edu.au /~roberto/teach/matrix/special.html (2558 words)

	Matrix Market: Glossary
		A defective matrix cannot be transformed to a diagonal matrix using similarity transformations.
		A matrix is diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row (or column).
		A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal).
	math.nist.gov /MatrixMarket/glossary.html (756 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)

	Each Matrix Is Equivalent to an Extended Diagonal Matrix
		An extended diagonal matrix is defined to be the matrix obtained by taking a diagonal matrix, adding enough zero rows at the bottom and enough zero columns on the right so that it becomes a matrix of the desired size.
		Each matrix is equivalent to an extended diagonal matrix whose upper diagonal entries are divisors of lower diagonal entries.
		may be transfigured to an extended diagonal matrix with the specified property by a finite number of elementary or secondary operations, i.e.
	amath.nchu.edu.tw /~hsu/run/node78.html (618 words)

	GLOSSARY (Site not responding. Last check: 2007-11-07)
		A band matrix is one whose entries are all zero except for the diagonal and a few of the immediately adjacent diagonals, whose entries may be nonzero.
		The extreme cases are a full matrix (all entries nonzero) which is a single submatrix, and a diagonal matrix, which consists of N submatrices, each diagonal element.
		The transpose of a matrix is the matrix obtained by switching the pairs of entries A(I,J) and A(J,I).
	www.psc.edu /general/software/packages/eispack/glossary.html (988 words)

	No Title
		Remember that diagonalizing a matrix requires finding a basis with respect to which the matrix is diagonal.
		In the general case, the diagonal blocks are ``nearly diagonal,'' and J is block diagonal.
		The eigenvalues of this upper triangular matrix are on the diagonal.
	www-math.cudenver.edu /~wbriggs/5718s01/notes_jordan/notes_jordan.html (1279 words)

	Linear Algebra (Site not responding. Last check: 2007-11-07)
		Informally, the transpose of a square matrix is obtained by reflecting at the main diagonal (that runs from the top left to bottom right of the matrix).
		Formally, the transpose of the m-by-n matrix A is the n-by-m matrix A
		The matrix B is called the inverse matrix of A. A common notation for the inverse of a matrix A is A
	linear-algebra.org /5.html (547 words)

	The diagonal matrix (Site not responding. Last check: 2007-11-07)
		The diagonal matrix is yet another initialization method, often used where the values can be stored more compact as a vector of data than as a full matrix.
		Diagonal matrices are also used because the original data were present as a vector.
		When the diagonal matrix is assigned, all off-diagonal elements in the submatrix are zero.
	www.fou.uib.no /fd/1996/h/413002/node44.html (193 words)

	Computation of the Rotation Matrix
		In motion analysis, it is often necessary to compute the transformation matrix directly from the coordinates of the markers fixed on a moving body.
		Once the transformation matrix is known, the orientation angles and eventually the location of center of rotation as well may be computed.
		In addition, matrix c' must be orthogonal because all three matrices composing c' are orthogonal.
	kwon3d.com /theory/jkinem/rotmat.html (415 words)

	Matrix and Quaternion FAQ (Site not responding. Last check: 2007-11-07)
		A matrix with M rows and N columns is defined as a MxN matrix.
		The inverse of an identity matrix is the identity matrix.
		In this case, the value of each matrix element is defined by: -1 1 i+j M = ----- * det submat M * -1 j,i det M i,j Q22.
	www.flipcode.com /documents/matrfaq.html (7478 words)

	Guide and Reference (Site not responding. Last check: 2007-11-07)
		A sparse matrix is a matrix having a relatively small number of nonzero elements.
		The diagonal-out skyline storage mode used for sparse matrices has two variations, depending on whether the matrix is a general sparse matrix or a symmetric sparse matrix.
		The profile-in skyline storage mode used for sparse matrices has two variations, depending on whether the matrix is a general sparse matrix or a symmetric sparse matrix.
	csit1cwe.fsu.edu /extra_link/essl/essl148.html (2642 words)

	Chapter 5. Testing for Equivalence (Site not responding. Last check: 2007-11-07)
		The computation of the kxk minors of a matrix can be a long task, not only because the computation of each determinant is a bit of a job, but also because there can be many of them.
		This command returns a matrix with one row whose entries are the kxk minors of the mxn matrix A, whenever k is a positive integer that is not greater than m and not greater than n.
		Another method of finding the strongly diagonal matrix D equivalent to A is to use `minors' to find the product of the first k diagonal entries of D, as in the proof in Section 3.
	www.nyu.edu /classes/edwards/chap5.html (350 words)

	Geometric Functional Analysis Seminar Abstract (Site not responding. Last check: 2007-11-07)
		A secular equation for the eigenvalues of a diagonal matrix perturbation
		We shall discuss how the eigenvalues of a diagonal complex matrix change when the matrix is perturbed by a matrix of rank one, of rank two, and so on.
		They also show that a matrix with distinct eigenvalues can be perturbed by a rank-one matrix to a matrix with any prescribed eigenvalue list.
	www.math.psu.edu /gfa/PastSeminars/SP00Abstracts/secular.html (112 words)

	Gauss-Jordan Elimination
		Again, we are transforming the coefficient matrix into another matrix that is much easier to solve, and the system represented by the new augmented matrix has the same solution set as the original system of linear equations.
		In Gauss-Jordan Elimination, the goal is to transform the coefficient matrix into a diagonal matrix, and the zeros are introduced into the matrix one column at a time.
		Notice that the coefficient matrix is now a diagonal matrix with ones on the diagonal.
	ceee.rice.edu /Books/CS/chapter2/linear44.html (439 words)

	Physics Help and Math Help - Physics Forums - is this a diagonal matrix
		diagonal matrix is a matrix with each element is0 excep for elements on the major diagonal.
		12-26-2003 03:07 PM A (square) matrix is diagonal iff the nondiagonal entries are zero.
		The diagonal entries are permitted to be zero as well, so this matrix is diagonal, with diagonal entires 2, 0, and 3:
	www.physicsforums.com /printthread.php?t=11599 (188 words)

	TriDiagonalSystems.html (Site not responding. Last check: 2007-11-07)
		There are at least two basic approaches, one of which is a more standard matrix manipulation technique where the tri-diagonal matrix is decomposed into a lower and an upper diagonal form.
		The other approach is an iterative approximation technique that less robust, it assumes the matrix is diagonal dominnat, but it is also perhaps a simpler algorithm for implementation on an FPGA.
		Let us generate a tri-diagonal matrix M such that the diagonal elements are drawn at random from a uniform random number generator in the range 0.1 to 1.0, and the off diagonal elements are drawn from the range 0.0 to 0.1.
	www.cs.colostate.edu /~ross/research/TriDiagonalSystems/TriDiagonalSystems1.html (361 words)

	Matrix Operations, Part 3
		A square matrix with ones on the main diagonal and zeros everywhere else is called an identity matrix.
		A matrix of any size with all zero entries is called a zero matrix.
		A matrix with zeros everywhere except possibly on the main diagonal in called a diagonal matrix.
	www.math.duke.edu /education/ccp/materials/diffeq/matrix/matr3.html (256 words)

	Constant Diagonal Matrix (DSP Blockset) (Site not responding. Last check: 2007-11-07)
		The Constant Diagonal Matrix block outputs a square diagonal matrix constant.
		The Number of rows in matrix parameter sets the matrix size, and the Constant along diagonal parameter determines the values along the matrix diagonal.
		If the vector specified for the Constant along diagonal parameter is larger than the specified number of rows in the matrix, the extra values at the end of the vector are ignored.
	www.math.muni.cz /matlab/toolbox/dspblks/constantdiagonalmatrix.html (148 words)

	Sparce Matrix Method
		This knowledge allows one to calculate the normal matrix as a sparse matrix, the vast majority of the off-diagonal elements are never calculated or even have computer memory allocated for their storage.
		The only elements calculated are the diagonal ones (including contributions from both the crystallographic and stereochemical restraints) and the off-diagonal elements for parameters from atoms directly connected by geometric restraints.
		Even with the simplification of the normal matrix introduced by the sparse approximation the problem of inverting the matrix is difficult.
	www.uoxray.uoregon.edu /dale/papers/CCP4_1994/node6.html (536 words)

	Full, Tridiagonal and Diagonal Matrix Interaction (Site not responding. Last check: 2007-11-07)
		The tridiagonal and the diagonal approximation of the POP self-energy predict a smaller current than the full matrix calculation, by factors of 3.1-3.7 and 8.3-10.6, respectively.
		Also, the calculated electron density is reduced by about the same factors, as shown for a single bias voltage of 0.44 V in Figure (2b).
		Figure 2: (a) Valley current calculated for three different POP models: full matrix (solid line), tridiagonal matrix (dashed line), and diagonal matrix (dotted line), and experimental data (long dashed line).
	www.cfdrc.com /nemo/pubs/alexandria_html/node3.html (191 words)

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