3,289 research outputs found
A note on a relationship between the inverse eigenvalue problems for nonnegative and doubly stochastic matrices and some applications
In this note, we establish some connection between the nonnegative inverse
eigenvalue problem and that of doubly stochastic one. More precisely, we prove
that if is the spectrum of an
nonnegative matrix A with Perron eigenvalue r, then there exists a least real
number such that is
the spectrum of an nonnegative generalized doubly stochastic matrix
for all As a consequence, any solutions for the nonnegative
inverse eigenvalue problem will yield solutions to the doubly stochastic
inverse eigenvalue problem. In addition, we give a new sufficient condition for
a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in
this case B is shown to be the unique closest doubly stochastic matrix to A
with respect to the Frobenius norm. Some related results are also discussed.Comment: 8 page
On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
We consider the distributions of the lengths of the longest weakly increasing
and strongly decreasing subsequences in words of length N from an alphabet of k
letters. We find Toeplitz determinant representations for the exponential
generating functions (on N) of these distribution functions and show that they
are expressible in terms of solutions of Painlev\'e V equations. We show
further that in the weakly increasing case the generating function gives the
distribution of the smallest eigenvalue in the k x k Laguerre random matrix
ensemble and that the distribution itself has, after centering and normalizing,
an N -> infinity limit which is equal to the distribution function for the
largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices
of trace zero.Comment: 30 pages, revised version corrects an error in the statement of
Theorem
Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems
Recently, we used the Sinc collocation method with the double exponential
transformation to compute eigenvalues for singular Sturm-Liouville problems. In
this work, we show that the computation complexity of the eigenvalues of such a
differential eigenvalue problem can be considerably reduced when its operator
commutes with the parity operator. In this case, the matrices resulting from
the Sinc collocation method are centrosymmetric. Utilizing well known
properties of centrosymmetric matrices, we transform the problem of solving one
large eigensystem into solving two smaller eigensystems. We show that only
1/(N+1) of all components need to be computed and stored in order to obtain all
eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We
applied our result to the Schr\"odinger equation with the anharmonic potential
and the numerical results section clearly illustrates the substantial gain in
efficiency and accuracy when using the proposed algorithm.Comment: 11 pages, 4 figure
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