Mathematics – Spectral Theory
Scientific paper
2006-07-28
Duke Math. J. 143 (2008), 205--223
Mathematics
Spectral Theory
Final version; 13 pages, no figures, LaTeX2e
Scientific paper
For $n \times n$ matrices $A$ and $B$ define $$\eta(A,B)=\sum_{S}\det(A[S])\det(B[S']),$$ where the summation is over all subsets of $\{1,..., n\}$, $S'$ 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 $A\geq 0$ and $B$ is Hermitian then (1) the polynomial $\eta(zA,-B)$ 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 $\eta(zA[\{i\}'],-B[\{i\}'])$ interlace those of $\eta(zA,-B)$. Assertions (1)-(3) solve three important conjectures proposed by C. R. Johnson 20 years ago. 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 that considerably generalize a certain subset of the Hadamard-Fischer-Koteljanskii inequalities for principal minors of positive definite matrices. Finally, we propose Lax type problems for real stable polynomials and mixed determinants.
Borcea Julius
Brändén Petter
No associations
LandOfFree
Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-365914