Principal minor sums of \((A + tB)^m\). (English) Zbl 1086.15506

Johnson, Charles R.; Leichenauer, Stefan; McNamara, Peter; Costas, Roberto

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.

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

Citations:

Zbl 0976.82501

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: 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