Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

Abstract

For ($n\times n$)-matrices $A$ and $B$, define ${\displaystyle \eta (A,B)=\sum _S\det (A[S])\det (B[S^{\text{'}}]),}$ where the summation is over all subsets of $\{1,\dots ,n\}$, $S^{\text{'}}$ is the complement of $S$, and $A[S]$ is the principal submatrix of $A$ with rows and columns indexed by $S$. We prove that if $A\ge 0$ and $B$ is Hermitian, then

(1) the polynomial $\eta (\mathrm{zA},-B)$ has all real roots;

(2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of $B$ if $A>0$; and

(3) for $1\le i\le n$, the roots of $\eta (\mathrm{zA}[\{i{\}}^{\text{'}}],-B[\{i{\}}^{\text{'}}])$ interlace those of $\eta (\mathrm{zA},-B)$.

Assertions (1)–(3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard, Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants