Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices

AbstractWe derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum. Moreover, we prove that any set λ1⩾λ2⩾ ⋯⩾λn is the spectrum of real symmetric centrosymmetric matrices S1 and S2 such that for S1 an eigenvector corresponding to λ1 is the all ones vector and for S2 an eigenvector corresponding to λ1 is the vector with components 1 and −1 alternately. The proof is constructive. Then, we derive an improved condition on {λk}k=1n such that S1=λ1E, where E is a stochastic symmetric centrosymmetric matrix. Finally, we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices. All the numerical procedures are based on the use of the Fast Fourier Transform

On an optimization problem of Bellman

AbstractThis paper considers a problem proposed by Bellman in 1970: given a continuous kernel K(x, y) defined on I × I, find a pair of continuous functions f and g such that f(x) + g(y) ⩾ K(x, y) on I × I and ∝I (f + g) is minimum. The notion of basic decomposition of K is defined, and it is shown that whenever K(x, y) or K(x, a + b − y), I = [a, b], admits a basic decomposition, Bellman's problem has a unique differentiable solution, provided K is differentiable. Explicit formulas for such solutions are given. More generally, there are kernels which admit basic decompositions on subintervals which can be “pasted together” to define a unique piecewise differentiable solution

A decreasing sequence of eigenvalue localization regions

AbstractWe construct a decreasing sequence of rectangles (Rp) in such a way that all the eigenvalues of a complex matrix A are contained in each rectangle. When A is a matrix with real spectrum or a normal matrix, each Rp can be obtained without knowing the eigenvalues of A

The spectra of the adjacency matrix and Laplacian matrix for some balanced trees

AbstractLet T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let dk−j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of T. They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k×k. The codiagonal entries for both matrices are dj-1,2⩽j⩽k-1, and dk, while the diagonal entries are zeros, in the case of the adjacency matrix, and dj, 1⩽j⩽k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues

Spectra of generalized Bethe trees attached to a path

AbstractA generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Pm be a path of m vertices. Let {Bi:1⩽i⩽m} be a set of generalized Bethe trees. Let Pm{Bi:1⩽i⩽m} be the tree obtained from Pm and the trees B1,B2,…,Bm by identifying the root vertex of Bi with the i-th vertex of Pm. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of Pm{Bi:1⩽i⩽m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B1=B2=…=Bm

A sufficient condition for the differentiable optimal solution for a problem of Bellman

AbstractIn this paper, we present a sufficient condition for the differentiable optimal solution for a problem proposed by R. Bellman [Bull. Amer. Math. Soc. 76 (1970), 971]: given a continuous kernel K(x, y) defined on I × I, I = [a, b], find a pair of continuous functions ƒ and g such that ƒ(x) + g(y) ⩾ K(x, y) on I × I and ∝I(ƒ + g) is minimum. More precisely, we prove that if K is of class C(2) and Kxy does not change sign on I × I then Bellman's problem has a unique differentiable solution which can be found by integrating a pair of differential equations