Johnson, Charles R.; Leichenauer, Stefan; McNamara, Peter; Costas, Roberto Principal minor sums of \((A + tB)^m\). (English) Zbl 1086.15506 Linear Algebra Appl. 411, 386-389 (2005). Summary: The question is raised whether the sum of the \(k \times k\) principal minors of the titled matrix is a polynomial (in \(t\)) with positive coefficients, when \(A\) and \(B\) are positive definite. This would generalize a conjecture made by D. Bessis, P. Moussa, and M. Villani [J. Math. Phys. 16, 2318–2325 (1975; Zbl 0976.82501)], as stated by E. H. Lieb and R. Seiringer [Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys. 115, 185–190 (2004)]. We give a variety of evidence for this further question, some of which is new. Cited in 3 Documents MSC: 15A15 Determinants, permanents, traces, other special matrix functions 15B57 Hermitian, skew-Hermitian, and related matrices 15A90 Applications of matrix theory to physics (MSC2000) Keywords:positive definite matrix; principal minors PDF BibTeX XML Cite \textit{C. R. Johnson} et al., Linear Algebra Appl. 411, 386--389 (2005; Zbl 1086.15506) Full Text: DOI References: [1] Bessis, D.; Moussa, P.; Villani, M., Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. math. phys., 16, 2318-2325, (1975) · Zbl 0976.82501 [2] Johnson, C.R.; Hillar, C., Eigenvalues of words in two positive definite letters, SIAM J. matrix anal. appl., 23, 916-928, (2002) · Zbl 1007.68139 [3] Hillar, C.; Johnson, C.R., On the positivity of the coefficients of a certain polynomial defined by two positive definite matrices, J. stat. phys., 118, 781-789, (2005) · Zbl 1126.15303 [4] Horn, R.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press New York · Zbl 0576.15001 [5] Lieb, E.H.; Seiringer, R., Equivalent forms of the bessis-moussa-villani conjecture, J. stat. phys., 115, 185-190, (2004) · Zbl 1157.81313 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.