On the Ilmonen-Haukkanen-Merikoski Conjecture (original) (raw)
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Linear Algebra and its Applications
Let K n be the set of all n × n lower triangular (0, 1)-matrices with each diagonal element equal to 1, L n = {Y Y T : Y ∈ K n } and let c n be the minimum of the smallest eigenvalue of Y Y T as Y goes through K n. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that c n is equal to the smallest eigenvalue of Y 0 Y T 0 , where Y 0 ∈ K n with (Y 0) ij = 1−(−1) i+j 2 for i > j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.
Interlacing properties of the eigenvalues of some matrix classes
Linear Algebra and its Applications, 2013
We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any its principal submatrix) for the class of matrices, introduced by Kotelyansky (all principal and all almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices also possess this property. We prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.
On the joint spectral radius of nonnegative matrices
Linear Algebra and its Applications
A 1. .. A n) i,j , where D ×D is the dimension of the matrices, U, V are respectively the largest entry and the smallest entry over all the positive entries of the matrices in A, and C is taken over all components in the dependency graph. The dependency graph is a directed graph where the vertices are the dimensions and there is an edge from i to j if and only if A i,j = 0 for some matrix A ∈ A. Furthermore, a bound on the norm is also given: There exists a nonnegative integer r so that for every n, const n r ρ(A) n ≤ max A1,...,An∈A A 1. .. A n ≤ const n r ρ(A) n. Corollaries of the approach include a simple proof for the joint spectral theorem for finite sets of nonnegative matrices and the convergence rate of some sequences. The method in use is mostly based on Fekete's lemma.
Inequalities Involving Lower-Triangular Matrices
Proceedings of the London Mathematical Society, 1980
The constant at the right of (1) was fixed by Landau [14] who showed that it is the best possible for each p. Since then many alternative proofs of Hardy's inequality have been given (cf. Broadbent [1], Elliot [5], Grandjot [7], Knopp [13], Kaluza and Szego [12]). Copson [2] generalized Theorem A by replacing the arithmetic mean of sequence u by a weighted arithmetic mean. We shall consider Copson's generalization in § 10. If A = {a mn) is the Cesaro matrix a mn = rnr 1 , where m ^ n, then Hardy's inequality can be written as (2) for 1 < p < oo. Petersen [18] and subsequently Davies and Petersen [4] produced sufficient conditions on a matrix A and an auxiliary sequence/ £ co for the existence of an inequality of the form \\A\x\\\ p^K \\f.a.x\\ p (l^p<co) for some K with a = {a mm }, the main diagonal sequence of A. Our object in this paper is to study inequalities of the form or, indeed, of the form (3) M i z i u , with (A, || • || A), (/x, || • 11^) being quasinormed FK spaces satisfying certain hypotheses. Henceforward, inequalities of the form (3) will* be referred to as HPD inequalities (for Hardy, Petersen, and Davies). Particular attention will be paid to the existence of best possible HPD inequalities, for fixed A, (A, ||*|| A), and (/x, || 1 1^); best possible not by virtue of the smallness of the number K, but of the sequence b, with respect to a particular definition of 'smallness' of sequences (see § 3). It is clear that the smaller the K, the better is the inequality (3). Just so, the 'smaller' the 6, the better is the inequality (3). We shall obtain conditions on A } (^>II'IIA)> a n d 0*ilHI/») for * n © existence of a 'smallest' b for which (3) holds for some K and will apply our results with A,/u among the l r and A specified or confined to some special class of matrices.
On the second real eigenvalue of nonegative and Z-matrices
Linear Algebra and its Applications, 1997
We give bounds for the second real eigenvalue of nonegative matrices and Z-matrices. Furthermore, we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices. Using these bounds, we prove that our inequality for the second real eigenvalue of the adjacency matrix of a connected regular graph improves a well-known bound for the second eigenvalue using Cheeger's inequality.
On the second real eigenvalue of nonnegative and Z-matrices
1999
We give bounds for the second real eigenvalue of nonnegative matrices and Z{matrices. Furthermore, we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices. Using this bounds we prove that our inequality for the second real eigenvalue of the incidence matrix of a connected regular graph improves a well known bound for the second eigenvalue using Cheeger's inequality.