Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A\in {ℝ}^{m\times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $\mathrm{Tr}\left(A^{-1}\right)$ and $\mathrm{Tr}\left(A^{-2}\right)$ have attracted attention recently, because they can be computed in $O\left(m\right)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed...

We consider the two-sided eigenproblem $A\otimes x=\lambda \otimes B\otimes x$ over max algebra. It is shown that any finite system of real intervals and points can be represented as spectrum of this eigenproblem.

Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.

Let $G=(V,E)$, $V=\{v_1,v_2,...,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^{+}=D+A$ and the normalized signless Laplacian matrix as ${ℒ}^{+}=D^{-1/2}{L}^{+}{D}^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r\left(G\right)={\gamma }_2^{+}/{\gamma }_n^{+}$ and $l\left(G\right)={\gamma }_2^{+}-{\gamma }_n^{+}$, where ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$ are eigenvalues of ${ℒ}^{+}$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads...

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $ℒ\left(T\right(a,b,c\left)\right)$ except $ℒ\left(T\right(t,t,2t+1\left)\right)$ ($t\ge 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $ℒ\left(T\right(t,t,2t+1\left)\right)$. Let ${\mu }_1\left(G\right)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of ${\mu }_1\left(ℒ\left(T(a,b,c)\right)\right)$, too.

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A\otimes x=\lambda \otimes x$ for usual eigenvectors. Under a suitable assumption,...