JAM
Journal of Applied Mathematics
16870042
1110757X
Hindawi Publishing Corporation
10.1155/2014/518487
518487
Editorial
Advances in Matrices, Finite and Infinite, with Applications 2014
Shivakumar
P. N.
^{1}
Psarrakos
Panayiotis J.
^{2}
Sivakumar
K. C.
^{3}
Zhang
Yang
^{1}
http://orcid.org/0000000177424416
da Fonseca
Carlos M.
^{4}
^{1}
Department of Mathematics, University of Manitoba, Winnipeg, MB
Canada
R3T 2N2
umanitoba.ca
^{2}
Department of Mathematics, School of Applied Mathematical & Physical Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens
Greece
ntua.gr
^{3}
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036
India
iitm.ac.in
^{4}
Department of Mathematics, Kuwait University, 13060 Safat
Kuwait
kuniv.edu
2014
7
9
2014
2014
19
08
2014
19
08
2014
7
9
2014
2014
Copyright © 2014 P. N. Shivakumar et al.
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Matrix theory either finite or infinite has increasingly proved to be a key element in many different modern scientific fields, far beyond its natural mathematical environment. This volume exposes the growing sophistication of the techniques involving matrices as well as some of many applications.
Classification of modular Lie algebras has recently been the subject of many authors. In particular, alteration techniques play a major role. Natural filtration for Cartan types has been shown to be invariant in infinite dimensional case. In this issue, Q. Mu in “Natural filtrations of infinitedimensional modular contact superalgebras” establishes that the natural filtration is invariant under automorphisms in the case of Lie algebras. The author uses adnilpotent elements techniques.
It is well known that rank of matrix plays an important role in matrix theory and has many applications in other areas. In the paper “Some results on characterizations of matrix partial orderings” by H. Wang and J. Xu, five matrix partial orderings defined in
C
m
×
n
are considered and the (left/right and both sided) star and sharp partial orderings are characterized by applying rank equalities.
Comparison theorems between the spectral radii of different matrices are useful tools for judging the efficiency of preconditioned. In S.X. Miao and Y. Cao’s paper “On comparison theorems for splittings of different semimonotone matrices,” the authors gave some comparisons for the spectral radii of matrices arising from proper splittings of different semimonotone matrices.
In the work reported by G. M. L Gladwell et al. in the paper “A test matrix for an inverse eigenvalue problem,” a simple derivation of a set of explicit expressions is provided for the components of a Jacobi matrix of order
n
×
n
. This matrix has the property that its eigenvalues come from the set
{
0,2
,
…
,
2
n

2
}
while also satisfying the additional condition that the eigenvalues of the leading principal
(
n

1
)
×
(
n

1
)
submatrix belong to the set
{
1,3
,
…
,
2
n

3
}
. As an application, an explicit solution for a springmass inverse problem is presented.
The combined matrix of a nonsingular matrix
A
is defined as
C
(
A
)
=
A
∘
(
A

1
)
T
, where
∘
means the Hadamard product. Combined matrices appear in the chemical literature, where they represent the relative gain array. Furthermore, the study of combined matrices yields the relation between the eigenvalues and diagonal entries of a diagonalizable matrix. It is well known that the row and column sums of a combined matrix are always equal to one, and, consequently, if
C
(
A
)
is entrywise nonnegative, then it has interesting properties and applications since it is a doubly stochastic matrix. In the paper “Nonnegative combined matrices” of R. Bru et al., the matrices which have nonnegative combined matrices are investigated. In particular, combined matrices of different classes of matrices, such as totally positive and totally negative matrices, and also when
A
is totally nonnegative and totally nonpositive, are studied.
P. N. Shivakumar
Panayiotis J. Psarrakos
K. C. Sivakumar
Yang Zhang
Carlos M. da Fonseca