arXiv subjects

C. R. Johnson

Publications and source records attributed to C. R. Johnson.

5 recordsLinked to original sources

arXiv2026

The INIEP: Irreducible and Positive Realizations

Our focus is upon {\it irreducible} nonnegative $n$-by-$n$ matrix realizations of nonnegatively realizable spectra or, equivalently, characteristic polynomials. After giving some general background, we make some useful new observations and show the existence of irreducible nonnegative realizations in some general cases. Then, we focus on $n<5$, where the NIEP is solved. Finally, we focus on the trace 0 case and, using graph theoretic methods, characterize nonnegative irreducible realizability among realizable polynomials. The closely related problem of positive realizations, for trace positive spectra, is also discussed.

math.CO↗

arXiv2024

Analysis of natural cardinal ranking vectors for pairwise comparisons and the universal efficiency of the Perron geometric mean

In models using pair-wise (ratio) comparisons among alternatives, a cardinal ranking vector should be deduced from a reciprocal matrix. The right Perron eigenvector (RP) was traditionally used, though several other options have emerged. We consider some alternatives, mostly new, namely the entry-wise reciprocal of the left Perron vector (LP), the left singular vector (LS), the entry-wise reciprocal of the right singular vector (RS), the arithmetic and geometric means of RP and LP (AP and GP), and of LS and RS (AS and GS). (The ranking vector AS was proposed by Gass and Rapcsák (2004)). All 8 of these vectors produce the natural vector in the consistent case. We compare them empirically, in terms of efficiency, for random matrices, as a function of the number of alternatives. It turns out that the vector GP is universily efficient, and this fact is proven. The vector GS performs better that the remaining 6 vectors. We show that, for reciprocal matrices obtained from consistent matrices by modifying one column and the corresponding row, all 8 vectors are efficient. Moreover, the cone generated by the columns is efficient.

math.CO↗

arXiv2018

Spectra of Tridiagonal Matrices over a Field

We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and sufficient that the necessarily real eigenvalues are distinct and those of the principal submatrix strictly interlace. So this case is very restrictive. By contrast, for a general field, the requirements on the two spectra are much less restrictive. In particular, in the real or complex case, the $n$-by-$n$ characteristic polynomial is arbitrary (so that the algebraic multiplicities may be anything in place of all 1's in the classical cases) and that of the principal submatrix is the complement of a lower dimensional algebraic set (and so relatively free). Explicit conditions are given.

math.CA↗

arXiv2016

Row Cones, Perron Similarities, and Nonnegative Spectra

In further pursuit of the diagonalizable \emph{real nonnegative inverse eigenvalue problem} (RNIEP), we study the relationship between the \emph{row cone} $\mathcal{C}_r(S)$ and the \emph{spectracone} $\mathcal{C}(S)$ of a Perron similarity $S$. In the process, a new kind of matrix, \emph{row Hadamard conic} (RHC), is defined and related to the D-RNIEP. Characterizations are given when $\mathcal{C}_r(S) = \mathcal{C}(S)$, and explicit examples are given for all possible set-theoretic relationships between the two cones. The symmetric NIEP is the special case of the D-RNIEP in which the Perron similarity $S$ is also orthogonal.

math.SP↗

arXiv2015

Matrices Totally Positive Relative to a Tree, II

In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed according to $T$.

math.CO↗