The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver...

We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our...

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non-universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and...

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the...

In this paper, we are concerned with the large $n$ limit of the distributions of linear combinations of the entries of a Brownian motion on the group of $n\times n$ unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic...

Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵\left(𝒢\right)$ be the center of $𝒢$. Suppose that $𝔮:𝒢\times 𝒢\to 𝒢$ is an $ℛ$-bilinear mapping and that ${𝔗}_{𝔮}:𝒢\to 𝒢$ is a trace of $𝔮$. The aim of this article is to describe the form of ${𝔗}_{𝔮}$ satisfying the centralizing condition $[{𝔗}_{𝔮}\left(x\right),x]\in 𝒵\left(𝒢\right)$ (and commuting condition $[{𝔗}_{𝔮}\left(x\right),x]=0$) for all $x\in 𝒢$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) ${𝔗}_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J\left(M_2\left(R\right)\right)$, or $I_2-A\in J\left(M_2\left(R\right)\right)$, or $A$ is similar to $\left(\begin{array}{cc}0& \lambda \\ 1& \mu \end{array}\right)$, where $\lambda \in J\left(R\right)$, $\mu \in 1+J\left(R\right)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J\left(R\right)$ and a root in $1+J\left(R\right)$. We further prove that $f\left(x\right)\in R\left[\right[x\left]\right]$ is strongly $J$-clean if $f\left(0\right)\in R$ be optimally $J$-clean.