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The bialternate matrix product revisited. (English) Zbl 1211.15009

Over the years, the bialternate matrix product, introduced by C. Stépanos [Journ. de Math. (5) 6, 73–128 (1900; JFM 31.0132.01)], was used as an efficient tool in stability theory, especially in studying polynomial and matrix stability and in the theory of dynamical systems in detecting and computing Hopf bifurcations in systems of ordinary differential equations.
In this paper, the bialternate product of two square matrices is re-examined together with another matrix product defined by means of the permanent function and having similar properties. The authors present old and new results concerning both products in a unified manner. A simple and elegant relation with the Kronecker product of matrices is also given.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus

Citations:

JFM 31.0132.01
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References:

[1] Bellman, R., Introduction to matrix analysis, (1960), McGraw-Hill Book Company Inc. · Zbl 0124.01001
[2] L. Elsner, Unpublished manuscript, 1998.
[3] Elsner, L.; Monov, V.; Szulc, T., On some properties of convex matrix sets characterized by P-matrices and block P-matrices, Linear and multilinear algebra, 50, 3, 199-218, (2002) · Zbl 1007.15012
[4] Fuller, A.T., Conditions for a matrix to have only characteristic roots with negative real parts, J. math. anal. appl., 23, 71-98, (1968) · Zbl 0157.15705
[5] Govaerts, W.; Sijnave, B., Matrix manifolds and the Jordan structure of the bialternate matrix product, Linear algebra appl., 292, 245-266, (1999) · Zbl 0936.15006
[6] Guckenheimer, J.; Myers, M.; Sturmfels, B., Computing Hopf bifurcations I, SIAM J. numer. anal., 34, 1-21, (1997) · Zbl 0948.37037
[7] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1995), Cambridge University Press
[8] Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Allyn and Bacon Rock Leigh New Jersey · Zbl 0126.02404
[9] Stéphanos, C., Sur une extension du calcul des substitutions lineaires, J. math. pure appl., 6, 73-128, (1900) · JFM 31.0132.01
[10] Wedderburn, J.H.M., Lectures on matrices, vol. 17, (1934), Colloquium Publications, American Mathematical Society Providence Rhode Island · Zbl 0010.09904
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