Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices

This paper gives simple proofs of Sylvester (` = 2) and Frobenius (` = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for ` > 3 matrices and an application to generalized inverses are provided.This paper has been partially supported by Ministerio de Economia y Competitividad (Grant DGI MTM2013-43678P and Red de Excelencia MTM2015-68805-REDT). The author thanks the referees for their valuable suggestions.Thome, N. (2016). Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices. Aequationes Mathematicae. 90(5):951-960. https://doi.org/10.1007/s00010-016-0412-4S951960905Baksalary J.K., Kala R.: The matrix equation AX − YB =  C. Linear Algebra Appl. 25, 41–43 (1979)Baksalary O.M., Trenkler G.: On k-potent matrices.. Electr. J. Linear Algebra 26, 446–470 (2013)Ben-Israel, A., Greville T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Wiley, New York (2003)Marsaglia G., Styan G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)Mosić D., Djordjević D.S.: Condition number of the W-weighted Drazin inverse. Appl. Math. Computat. 203(1), 308–318 (2008)Puntanen S., Styan G.P.H., Isotalo J.: Matrix Tricks for Linear Statistical Models. Springer, Berlin (2011)Tian Y., Styan G.P.H.: A new rank formula for idempotent matrices with applications. Comment. Math. Univ. Carolinae 43(2), 379–384 (2002)Wang G., Wei Y., Qiao S.: Generalized Inverses: Theory and Computations. Science Press, Beijing (2004