1. CMB Online first
2. CMB Online first
 Koşan, Tamer; Sahinkaya, Serap; Zhou, Yiqiang

Additive maps on units of rings
Let $R$ be a ring. A map $f: R\rightarrow R$
is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$
of $R$.
Here a map $f: R\rightarrow R$ is called unitadditive if $f(u+v)=f(u)+f(v)$
for all units $u$ and $v$ of $R$. Motivated by a recent result
of Xu, Pei and Yi
showing that, for any field $F$, every
unitadditive map of ${\mathbb M}_n(F)$ is additive for all $n\ge
2$, this paper is about the question when every unitadditive
map of a ring is additive. It is proved that every unitadditive
map of a semilocal ring $R$ is additive if and only if either
$R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong
\mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal
ring $R$, every unitadditive map of ${\mathbb M}_n(R)$ is additive
for all $n\ge 2$. These results are further extended to rings
$R$ such that $R/J(R)$ is a direct product of exchange rings
with primitive factors Artinian. A unitadditive map $f$ of a
ring $R$ is called unithomomorphic if $f(uv)=f(u)f(v)$ for all
units $u,v$ of $R$. As an application, the question of when every
unithomomorphic map of a ring is an endomorphism is addressed.
Keywords:additive map, unit, 2sum property, semilocal ring, exchange ring with primitive factors Artinian Categories:15A99, 16U60, 16L30 

3. CMB 2017 (vol 60 pp. 561)
 Kurdyka, Krzysztof; Paunescu, Laurentiu

Nuij Type Pencils of Hyperbolic Polynomials
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.
Keywords:hyperbolic polynomial, stable polynomial, determinantal representa tion, symmetric Toeplitz matrix Categories:15A15, 30C10, 47A56 

4. CMB 2017 (vol 60 pp. 861)
 Wang, Long; CastroGonzalez, Nieves; Chen, Jianlong

Characterizations of Outer Generalized Inverses
Let $R$
be a ring and $b, c\in R$.
In this paper, we give some characterizations of the $(b,c)$inverse,
in terms of the direct sum decomposition, the annihilator and
the invertible elements.
Moreover, elements with equal $(b,c)$idempotents related to
their $(b, c)$inverses are characterized, and the reverse order
rule for the $(b,c)$inverse is considered.
Keywords:$(b, c)$inverse, $(b, c)$idempotent, regularity, imagekernel $(p, q)$inverse, ring Categories:15A09, 16U99 

5. CMB 2017 (vol 60 pp. 807)
 Liu, Zhongyun; Qin, Xiaorong; Wu, Nianci; Zhang, Yulin

The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=CS$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the GaussSeidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
GaussSeidel iteration, no matter whether the CSCS itself is
convergent or not.
Keywords:Hermitian positive definite, CSCS splitting, GaussSeidel splitting, iterative method, Toeplitz matrix Categories:15A23, 65F10, 65F15 

6. CMB 2016 (vol 60 pp. 613)
 Reichstein, Zinovy; Vistoli, Angelo

On the Dimension of the Locus of Determinantal Hypersurfaces
The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$tuple $A := (A_1, \dots, A_r)$ of $n \times n$matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$tuple of $n \times n$matrices in general position,
then up to conjugacy, there are only finitely many $r$tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r1)n^2 + 1$.
Keywords:determinantal hypersurface, matrix invariant, $q$binomial coefficient Categories:14M12, 15A22, 05A10 

7. CMB 2016 (vol 60 pp. 269)
 Chen, Jianlong; Zhu, Huihui; Patricio, Pedro; Zhang, Yulin

Characterizations and Representations of Core and Dual Core Inverses
In this
paper,
double commutativity and the reverse order law for the core inverse
are considered. Then, new characterizations of the MoorePenrose
inverse of a regular element are given by onesided invertibilities
in a ring. Furthermore, the characterizations and representations
of
the core and dual core inverses of a regular element are considered.
Keywords:regularities, group inverses, MoorePenrose inverses, core inverses, dual core inverses, Dedekindfinite rings Categories:15A09, 15A23 

8. CMB 2016 (vol 59 pp. 585)
 Lin, Minghua

A Determinantal Inequality Involving Partial Traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$norms:
\[
1+\\mathbf{A}\_p\ge \\tr_1 \mathbf{A}\_p+\\tr_2 \mathbf{A}\_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]
Keywords:determinantal inequality, partial trace, block matrix Categories:47B65, 15A45, 15A60 

9. CMB 2016 (vol 59 pp. 311)
 Ilten, Nathan; Teitler, Zach

Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant
$\det_3$ is $5$,
and the product rank of the $3 \times 3$ permanent
$\operatorname{perm}_3$
is $4$.
As a corollary, we obtain that the tensor rank of $\det_3$ is
$5$ and the tensor rank of $\operatorname{perm}_3$ is $4$.
We show moreover that the border product rank of $\operatorname{perm}_n$ is
larger than $n$ for any $n\geq 3$.
Keywords:product rank, tensor rank, determinant, permanent, Fano schemes Categories:15A21, 15A69, 14M12, 14N15 

10. CMB 2014 (vol 58 pp. 207)
 Moslehian, Mohammad Sal; Zamani, Ali

Exact and Approximate Operator Parallelism
Extending the notion of parallelism we introduce the concept of
approximate parallelism in normed spaces and then substantially
restrict ourselves to the setting of Hilbert space operators endowed
with the operator norm. We present several characterizations of the
exact and approximate operator parallelism in the algebra
$\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a
Hilbert space $\mathscr{H}$. Among other things, we investigate the
relationship between approximate parallelism and norm of inner
derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the
parallel elements of a $C^*$algebra by using states. Finally we
utilize the linking algebra to give some equivalence assertions
regarding parallel elements in a Hilbert $C^*$module.
Keywords:$C^*$algebra, approximate parallelism, operator parallelism, Hilbert $C^*$module Categories:47A30, 46L05, 46L08, 47B47, 15A60 

11. CMB 2012 (vol 57 pp. 25)
 Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung

Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.
Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities Categories:47A63, 15A45 

12. CMB 2010 (vol 54 pp. 237)
 Creedon, Leo; Gildea, Joe

The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.
Categories:16U60, 16S34, 20C05, 15A33 

13. CMB 2009 (vol 52 pp. 295)
 P{\l}otka, Krzysztof

On Functions Whose Graph is a Hamel Basis, II
We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, \emph{i.e.,} for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.
Keywords:Hamel basis, additive, Hamel functions Categories:26A21, 54C40, 15A03, 54C30 

14. CMB 2009 (vol 52 pp. 9)
 Chassé, Dominique; SaintAubin, Yvan

On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics
Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.
Keywords:symmetric group, representation theory, eigenvalue, statistical physics Categories:20B30, 20C30, 15A18, 82B20, 82B28 

15. CMB 2008 (vol 51 pp. 86)
16. CMB 2006 (vol 49 pp. 313)
17. CMB 2006 (vol 49 pp. 281)
 Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.

Correction to a Theorem on Total Positivity
A wellknown theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(z)$ generates a PF sequence.
Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur function Categories:15A48, 15A45, 15A57, 05E05 

18. CMB 2005 (vol 48 pp. 394)
 Đoković, D. Ž.; Szechtman, F.; Zhao, K.

Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group
of rank $m$ over an infinite field $F$ of characteristic different
from $2$. We show that any $n\times n$ symmetric matrix $A$ is
equivalent under symplectic congruence transformations to the
direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal
and $C$ tridiagonal. Since the $\Sp_n(F)$module of symmetric
$n\times n$ matrices over $F$ is isomorphic to the adjoint module
$\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in
$\sp_n(F)$ has a representative in the sum of $3m1$ root spaces,
which we explicitly determine.
Categories:11E39, 15A63, 17B20 

19. CMB 2005 (vol 48 pp. 267)
 Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.

Continuous Adjacency Preserving Maps on Real Matrices
It is proved that every adjacency preserving continuous map
on the vector space of real matrices of fixed size, is either a
bijective affine tranformation
of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if
the matrices are of square size, or its range is contained
in a linear subspace consisting of matrices of rank at most one
translated by some matrix $R$. The result
extends previously known
theorems where the map was assumed to be also injective.
Keywords:adjacency of matrices, continuous preservers, affine transformations Categories:15A03, 15A04. 

20. CMB 2004 (vol 47 pp. 73)
 Li, Ma; Dezhong, Chen

Systems of Hermitian Quadratic Forms
In this paper, we give some conditions to judge when a system of
Hermitian quadratic forms has a real linear combination which is
positive definite or positive semidefinite. We also study some
related geometric and topological properties of the moduli space.
Keywords:hermitian quadratic form, positive definite, positive semidefinite Category:15A63 

21. CMB 2003 (vol 46 pp. 332)
 Đoković, Dragomir Z.; Tam, TinYau

Some Questions about Semisimple Lie Groups Originating in Matrix Theory
We generalize the wellknown result that a square traceless complex
matrix is unitarily similar to a matrix with zero diagonal to
arbitrary connected semisimple complex Lie groups $G$ and their Lie
algebras $\mathfrak{g}$ under the action of a maximal compact subgroup
$K$ of $G$. We also introduce a natural partial order on
$\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for
all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This
partial order is $K$invariant and induces a partial order on the
orbit space $\mathfrak{g}/K$. We prove that, under some restrictions
on $\mathfrak{g}$, the set $f(K\cdot x)$ is starshaped with respect
to the origin.
Categories:15A45, 20G20, 22E60 

22. CMB 2003 (vol 46 pp. 54)
 Cheung, WaiShun; Li, ChiKwong

Linear Maps Transforming the Unitary Group
Let $U(n)$ be the group of $n\times n$ unitary matrices. We show that if
$\phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is
a multiple of $n$, and $\phi$ has the form
$$
A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W
$$
for some $V, W \in U(m)$. From this result, one easily deduces the
characterization of linear operators that map $U(n)$ into itself obtained
by Marcus. Further generalization of the main theorem is also discussed.
Keywords:linear map, unitary group, general linear group Category:15A04 

23. CMB 2001 (vol 44 pp. 270)
 Cheung, WaiShun; Li, ChiKwong

Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$.
The $c$numerical range of an $n \times n$ matrix $A$ is defined by
$$
W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\}
\text{ an orthonormal basis for } \IC^n \Bigr\},
$$
and the $c$numerical radius of $A$ is defined by $r_c (A) = \max
\{z : z \in W_c (A)\}$. We determine the structure of those linear
operators $\phi$ on algebras of block triangular matrices, satisfying
$$
W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or}
\quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.
$$
Keywords:linear operator, numerical range (radius), block triangular matrices Categories:15A04, 15A60, 47B49 

24. CMB 2000 (vol 43 pp. 448)
 Li, ChiKwong; Zaharia, Alexandru

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a
subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider
the {\it generalized matrix function} on $m\times m$ matrices $B =
(b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m
b_{j\sigma(j)}$ and the {\it generalized numerical range} of an
$n\times n$ complex matrix $A$ associated with $d^H$ defined by
$$
\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that }
X^*X = I_m\}.
$$
It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$.
We show that there exist normal matrices $A$ for which $\wmp(A)$ is
not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove
that a normal matrix $A $ with eigenvalues lying on a straight line
has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some
nonzero $\nu \in \IC$. These results extend those of Hu, Hurley
and Tam, who studied the special case when $2 \le m \le 3 \le n$
and $H = S_m$.
Keywords:convexity, generalized numerical range, matrices Category:15A60 

25. CMB 1998 (vol 41 pp. 178)