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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Positive splittings of matrices and their nonnegative Moore-Penrose inverses

100%
Kurmayya T., Sivakumar K.
Discussiones Mathematicae - General Algebra and Applications
|
2008
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tom 28
|
nr 2
227-235
EN
In this short note we study necessary and sufficient conditions for the nonnegativity of the Moore-Penrose inverse of a real matrix in terms of certain spectral property shared by all positive splittings of the given matrix.
2
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The vector cross product from an algebraic point of view

80%
Trenkler G.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 1
67-82
EN
The usual vector cross product of the three-dimensional Euclidian space is considered from an algebraic point of view. It is shown that many proofs, known from analytical geometry, can be distinctly simplified by using the matrix oriented approach. Moreover, by using the concept of generalized matrix inverse, we are able to facilitate the analysis of equations involving vector cross products.
3
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Elements of C*-algebras commuting with their Moore-Penrose inverse

80%
Koliha J. J.
Studia Mathematica
|
2000
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tom 139
|
nr 1
81-90
EN
We give new necessary and sufficient conditions for an element of a C*-algebra to commute with its Moore-Penrose inverse. We then study conditions which ensure that this property is preserved under multiplication. As a special case of our results we recover a recent theorem of Hartwig and Katz on EP matrices.
4
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Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space

80%
Appi Reddy K., Kurmayya T.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.
5
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On generalized inverses of singular matrix pencils

70%
Röbenack K., Reinschke K.
International Journal of Applied Mathematics and Computer Science
|
2011
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tom 21
|
nr 1
161-172
EN
Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.
6
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On melancholic magic squares

70%
Trenkler G., Trenkler D.
Discussiones Mathematicae Probability and Statistics
|
2013
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tom 33
|
nr 1-2
111-119
EN
Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.
7
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Application of the partitioning method to specific Toeplitz matrices

70%
Stanimirović P., Miladinović M., Stojanović I., Miljković S.
International Journal of Applied Mathematics and Computer Science
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2013
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tom 23
|
nr 4
809-821
EN
We propose an adaptation of the partitioning method for determination of the Moore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore-Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in MATLAB, and illustrative examples are presented.
8
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Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

61%
Kurata H., Bapat R. B.
Special Matrices
|
2016
|
tom 4
|
nr 1
270-282
EN
By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.
9
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A discrepancy principle for Tikhonov regularization with approximately specified data

61%
Thamban Nair M., Schock E.
Annales Polonici Mathematici
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1998
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tom 69
|
nr 3
197-205
EN
Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T + αI)x_α^δ = T*y^δ$, $|y - y^δ| ≤ δ$, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T*y^δ$ are approximated by Aₙ and $zₙ^δ$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).
10
Content available remote

Equalities for orthogonal projectors and their operations

51%
Tian Y.
Open Mathematics
|
2010
|
tom 8
|
nr 5
855-870
EN
A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.
11
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Note on the core matrix partial ordering

51%
Mielniczuk J.
Discussiones Mathematicae Probability and Statistics
|
2011
|
tom 31
|
nr 1-2
71-75
EN
Complementing the work of Baksalary and Trenkler [2], we announce some results characterizing the core matrix partial ordering.
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