Chen, Xi; Zheng, Sai-Nan A unified proof of interlacing properties of eigenvalues of totally positive matrices. (English) Zbl 1483.15009 Linear Algebra Appl. 632, 241-245 (2022). The manuscript presents a complete simple proof of interlacing of eigenvalues for the class of totally positive matrices, i.e., matrices with nonnegative minors. The proof is based on the characterizations of both interlacing and compatibility properties of polynomials with real zeros. Reviewer: Iveta Hnetynkova (Praha) MSC: 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors 26C10 Real polynomials: location of zeros 05A20 Combinatorial inequalities Keywords:polynomial with only real zeros; totally positive matrix; interlacing; compatibility PDFBibTeX XMLCite \textit{X. Chen} and \textit{S.-N. Zheng}, Linear Algebra Appl. 632, 241--245 (2022; Zbl 1483.15009) Full Text: DOI References: [1] Chudnovsky, M.; Seymour, P., The roots of the independence polynomials of a clawfree graph, J. Comb. Theory, Ser. B, 97, 350-357 (2007) · Zbl 1119.05075 [2] Fisk, S., Polynomials, roots and interlacing [3] Gantmacher, F. R.; Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (2002), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI · Zbl 1002.74002 [4] Gantmacher, F. R.; Krein, M. G., Sur les matrices complètement nonnégatives et oscillatoires, Compos. Math., 4, 445-476 (1937) · JFM 63.0038.04 [5] Haglund, J.; Zhang, P. B., Real-rootedness of variations of Eulerian polynomials, Adv. Appl. Math., 109, 38-54 (2019) · Zbl 1415.05016 [6] Koteljanskii, D. M., Some sufficient conditions for reality and simplicity of the spectrum of a matrix, Transl. Am. Math. Soc., 27, 35-42 (1963) · Zbl 0128.01807 [7] Li, C. K.; Mathias, R., Interlacing inequalities for totally nonnegative matrices, Linear Algebra Appl., 341, 35-44 (2002) · Zbl 0997.15014 [8] Liu, L. L.; Wang, Y., A unified approach to polynomoal sequences with only real zeros, Adv. Appl. Math., 38, 542-560 (2007) · Zbl 1123.05009 [9] Marcus, A. W.; Spielman, D. A.; Srivastava, N., Interlacing families I: bipartite Ramanujan graphs of all degrees, Ann. Math., 182, 307-325 (2015) · Zbl 1316.05066 [10] Obreschkoff, N., Verteilung und Berechnung der Nullstellen reeller Polynome (1963), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin, (German) · Zbl 0156.28202 [11] Pinkus, A., An interlacing property of eigenvalues of strictly totally positive matrices, Linear Algebra Appl., 279, 201-206 (1998) · Zbl 0933.15034 [12] Pinkus, A., Totally Positive Matrices (2010), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1185.15028 [13] Wang, Y.; Yeh, Y.-N., Polynomials with real zeros and Pólya frequency sequences, J. Comb. Theory, Ser. A, 109, 63-74 (2005) · Zbl 1057.05007 [14] Wang, Y.; Zheng, S.-N., The converse of Weyl’s eigenvalue inequality, Adv. Appl. Math., 109, 65-73 (2019) · Zbl 1416.15016 [15] Yang, A. L.B.; Zhang, P. B., Brenti’s open problem on the real-rootedness of q-Eulerian polynomials of type D, SIAM J. Discrete Math., 31, 918-926 (2017) · Zbl 1366.05008 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.