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Topic: Block diagonal matrix

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	Block matrix - Wikipedia, the free encyclopedia
		In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks.
		A block diagonal matrix is a block matrix which is a square matrix, and having main diagonal blocks square matrices, such that the off-diagonal blocks are zero matrices.
		In linear algebra terms, the use of a block matrix corresponds to having a linear mapping thought of in terms of corresponding 'bunches' of basis vectors.
	en.wikipedia.org /wiki/Block_matrix (630 words)

	Inverse of structured rank matrices
		matrix plus a block diagonal matrix for which the sizes of the blocks of the first and the latter diagonal are the same.
		In all the theorems the diagonal elements are excluded from the structure.
		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)

	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)

	NCLDUDecomposition[aMatrix, Options]
		, and the fourth is the permutation matrix
		If all the diagonal elements are zero, then a permutation matrix is used to place a nonzero entry in the (2,1) position which will automaticaly place a nonzero entry in the (1,2) position if the matrix is symmetric.
		block pivoting, returning instead, the remaining undecomposed block with zeros along the diagonal as a final block diagonal entry.
	math.ucsd.edu /~ncalg/NCBIGDOC02/node64.html (565 words)

	Matrix Manual: Special Matrices (Site not responding. Last check: 2007-11-07)
		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.psi.toronto.edu /matrix/special.html (2192 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)

	Representing a Class of Similar Matrices
		The blocks at the upper left, that come from k, are still simple jordan blocks, but now they have l down the main diagonal instead of 0.
		Every matrix m is similar to a jordan canonical matrix j, and j serves as the "canonical representative" for the class of matrices similar to m.
		The jordan blocks that are nilpotent are the ones with 0 on the main diagonal, i.e.
	www.mathreference.com /la-jf,rep.html (1240 words)

	subspace iteration in the similarity reduction
		As a consequence, depending on the gaps between the eigenvalues, the semiseparable part of the matrix will converge to a block diagonal matrix, and the eigenvalues of these blocks converge to the largest eigenvalues in absolute value of the original symmetric matrix.
		We will see that the reduction algorithm from a symmetric to a semiseparable matrix can be interpreted as such a kind of subspace iteration, where the dimension of the subspace grows by one at each step of the algorithm.
		This explains why in the numerical examples (see Chapter 6), the lower right block already gives a good estimate of the largest eigenvalues, since they are connected to a subspace on which the subspace iteration is performed most.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node58.html (1256 words)

	Contents of Transform Model File Format
		The first block represents the transformation for only the static components of the feature vector, while the second represents the deltas and the third the accelerations.
		This block diagonal matrix example makes the assumption that for the transformation, there is no correlation between the statics, deltas and delta deltas.
		The variance transformation is a diagonal matrix and as such is simply stored as a vector.
	www.icsi.berkeley.edu /Speech/docs/HTKBook/node129_ct.html (519 words)

	lapack-d/dspsvx.html (Site not responding. Last check: 2007-11-07)
		The factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, steps 3 and 4 are skipped.
		The system of equations is solved for X using the fac- tored form of A. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
		The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solu- tion and error bounds could not be computed.
	www.math.utah.edu:8080 /software/lapack/lapack-d/dspsvx.html (629 words)

	Notes on Diagonalizing Operators
		From these statements, we can see that multiplying the matrix times the transpose of the matrix of cofactors gives a diagonal matrix with the diagonal elements all equal to the determinant.
		The Hamiltonian in the symmetry eigenstates will then be block diagonal with the elements in each block having the same symmetry operator eigenvalue.
		If we reorder the states, this is block diagonal since state 1 only couples to state 4, and 2 couples to state 3.
	phyastweb.la.asu.edu /phy576-schmidt/diagonalization/index.html (1513 words)

	Linear Algebra - Preliminary lectures
		A block matrix is a matrix which can be divided into similar parts i.e.
		Since the matrix corresponding to the linear transform is diagonal and since each eigenvector corresponds to a dimension (i.e.
		is an orthogonal matrix is a matrix square root of
	vision.unige.ch /~marchand/teaching/linalg/linalg.html (1819 words)

	Epsilon Decomposition (Site not responding. Last check: 2007-11-07)
		Input matrix data have to be provided as csv ascii compressed file.
		The next step is to find what matrix rows are connected, then permute the matrix rows and columns in order to to get a block diagonal decomposed matrix.
		After permuting the original matrix one needs to remove elements that are outside the blocks in order to get a block diagonal decomposed matrix.
	www.epsdec.com /index.html (422 words)

	Block Diagonal Matrix (Site not responding. Last check: 2007-11-07)
		You may have noticed that every matrix is block diagonal, consisting of one block running from 1 to n.
		As a check, apply this result to a diagonal matrix, where each block is a single element.
		Similarly, the trace of the matrix is the sum of the traces of the individual blocks.
	www.mathreference.com /la-jf,bdiag.html (515 words)

	Software and Appendices for "Using Noncommutative Groebner bases in solving partially prescribed matrix inverse ...
		We consider block matrix completion problems similar to those in a paper in 1995 by W.W. Barrett, C.R. Johnson, M.E. Lundquist and H.J. Woerdeman, "Completing a Block Diagonal Matrix With a Partially Prescribed Inverse".
		The particular block matrix inverse completion problems we have studied are those which have 7 unknown blocks and 11 known blocks.
		Here we describe a general method by which all block matrix completion problems of this type may be analyzed with sufficient computational power.
	math.ucsd.edu /~ncalg/DELL/MatrixComplete/mtxcomplete.html (511 words)

	LU Decomposition
		A block tri-diagonal matrix (see the figure) frequently arises when solving multiple differential equations, either two-point boundary value problems or partial differential equations.
		When solving several simultaneous equations it is important to arrange the unknowns in a particular way to achieve the block tri-diagonal form of the matrix and the resulting efficiencies.
		In this example, suppose the block tri-diagonal matrix is composed of blocks that are n
	faculty.washington.edu /finlayso/ebook/algebraic/advanced/LUDecomp.htm (520 words)

	The main routine
		For reasons that will be explained later, if there are numerous small blocks each of dimension say less than 10, we group them together as a single sparse block-diagonal matrix instead of considering them as individual blocks.
		Notice that when the block is a diagonal matrix, only the diagonal elements are stored, and they are stored as a column vector.
		The reason for using the sparse matrix representation to handle the case when we have numerous small diagonal blocks is that it is less efficient for M
	www.math.cmu.edu /~reha/eguide/node13.html (1073 words)

	[No title]
		N (input) INTEGER The order of the matrix A. A (input/output) COMPLEX array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A con tains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
		On exit, the block diagonal matrix D and the mul tipliers used to obtain the factor U or L (see below for further details).
		The factorization has been completed, but the block diagonal matrix D is exactly singular, and divi sion by zero will occur if it is used to solve a system of equations.
	www.ibiblio.org /gferg/ldp/man/manl/csytrf.l.html (608 words)

	Constant Diagonal Matrix (DSP Blockset)
		The Constant Diagonal Matrix block outputs a square diagonal matrix constant.
		Choose Inherit from `Constant(s) along diagonal' to set the output data type and scaling to match the values of the Constant(s) along diagonal parameter.
		Matrix Operations -- List of all blocks in the Matrix Operations library.
	www.weizmann.ac.il /matlab/toolbox/dspblks/constantdiagonalmatrix.html (524 words)

	lapack-z/zsptrf.html
		N (input) INTEGER The order of the matrix A. AP (input/output) COMPLEX16 array, dimension (N(N+1)/2) On entry, the upper or lower triangle of the sym- metric matrix A, packed columnwise in a linear array.
		On exit, the block diagonal matrix D and the multi- pliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).
		The fac- torization has been completed, but the block diago- nal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
	www.math.utah.edu /software/lapack/lapack-z/zsptrf.html (408 words)

	block diagonal matrices in MATLAB?
		It's called "blkdiag", and it works by taking all of the input arguments, which are matrices, and placing them on the diagonal of a larger sparse matrix.
		The problem I have with this function however, is that it doesn't create a block diagonal matrix of arbitrary size--it only concatenates the numer of arguments you input.
		In my case, all of the blocks are identical (viz., they are the tribanded matrix, B[1,-2,1]), and I want to place N-1 of these blocks on the diagonal of the larger matrix.
	www.physicsforums.com /showthread.php?t=77645 (262 words)

	zsptrf(l): compute factorization of complex ... - Linux man page
		On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array.
		On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).
		The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
	www.die.net /doc/linux/man/manl/zsptrf.l.html (595 words)

	zbdimm - block diagonal format matrix-matrix multiply
		IBDIAG() integer array of length NBDIAG consisting of the corresponding diagonal offsets of the non-zero block diagonals of A in VAL.
		Lower triangular block diagonals have negative offsets, the main block diagonal has offset 0, and upper triangular block diagonals have positive offset.
		NBDIAG the number of non-zero block diagonals in A. LB dimension of dense blocks composing A. B() rectangular array with first dimension LDB.
	docs.sun.com /source/816-2461/sbdimm.html (410 words)

	Nilpotent matrix - Wikipedia, the free encyclopedia
		This is an example of a 4×4 nilpotent matrix (in fact, matrices of this form are called shift matrices).
		There is a classification theorem showing that this is typical: a nilpotent matrix is similar to a block matrix, with diagonal square blocks generalizing this type, and other blocks zero.
		This fact follows from the Jordan decomposition theorem, together with the result that every matrix similar to a nilpotent matrix is also nilpotent.
	en.wikipedia.org /wiki/Nilpotent_matrix (269 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		pdBlocked package:nlme R Documentation Positive-Definite Block Diagonal Matrix Description: This function is a constructor for the 'pdBlocked' class, representing a positive-definite block-diagonal matrix.
		Each block-diagonal element of the underlying matrix is itself a positive-definite matrix and is represented internally as an individual 'pdMat' object.
		It is used to obtain the levels for 'factors', which affect the dimensions and the row/column names of the underlying matrix.
	www.oulu.fi /tietohallinto/ohjeet/unix/R/library/nlme/help/pdBlocked (424 words)

	TDecompBK
		The Bunch-Kaufman diagonal pivoting method decomposes a real symmetric matrix A using A = UDU^T where U is a product of permutation and unit upper triangular matrices, U^T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
		On exit fU contains the block diagonal matrix D and the multipliers used to obtain the factor U, see above.
		Set the matrix to be decomposed, decomposition status is reset.
	root.cern.ch /root/html512/TDecompBK.html (353 words)

	NAG C Library, Mark 7 : f07mrc
		is a 1 by 1 pivot block and the
		is a 2 by 2 pivot block and the
		has 2 by 2 blocks, only the upper or lower triangle is stored, as specified by uplo.
	www.nag.co.uk /numeric/cl/manual/xhtml/F07/f07mrc.xml (477 words)

	matrix (Site not responding. Last check: 2007-11-07)
		The eigenvalues (and eigenvectors) of a matrix are computed in a several step process which begins with a Hessenberg decomposition, followed by a Schur decomposition, from which the eigenvalues are apparent.
		Form a block matrix of size M by N, with a copy of matrix A as each element.
		Return a new matrix formed by extracting extract the lower (`tril') or upper (`triu') triangular part of the matrix A, and setting all other elements to zero.
	octave.sourceforge.net /index/matrix.html (1267 words)

	Create Diagonal Matrix (DSP Blockset) (Site not responding. Last check: 2007-11-07)
		The Create Diagonal Matrix block creates a square matrix from the diagonal specified by the vector input.
		The elements of the length-M input vector are used to populate the diagonal of an M-by-M matrix output.
		If you expect to generate code for the Create Diagonal Matrix block using the Real-Time Workshop, you should ensure that inputs are contiguous in memory.
	www.tau.ac.il /cc/pages/docs/matlab/help/toolbox/dspblks/creatediagonalmatrix.html (68 words)

	Matrix Generator JORDAN (Site not responding. Last check: 2007-11-07)
		This is a Java applet which generates a Jordan block, that is, a bidiagonal matrix with constant diagonal and ones on the first upper diagonal.
		The Jordan normal form of a matrix is a block diagonal matrix whose diagonal blocks are Jordan blocks.
		The Matrix Market is a service of the Mathematical and Computational Sciences Division / Information Technology Laboratory / National Institute of Standards and Technology.
	math.nist.gov /MatrixMarket/data/MMDELI/jordan/jordan.html (120 words)

	linear7sol
		Show that the matrix in the previous problem, part a, is a root of its own characteristic polynomial.
		Show that the eigenvalues of an upper triangular matrix are the diagonal entries of the matrix.
		Give another factorization of the matrix in the previous problem, part b.
	www-math.cudenver.edu /~rrosterm/linear7sol/linear7sol.html (374 words)

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