Maximum Principles for Normal Matrices

Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

Cite this paper

Dax, A. (2019) Maximum Principles for Normal Matrices. Advances in Linear Algebra & Matrix Theory, 9, 73-81. doi: 10.4236/alamt.2019.93005.

 [1] Bhatia, R. (1997) Matrix Analysis. Springer, New York. https://doi.org/10.1007/978-1-4612-0653-8 [2] Carlson, D. (1983) Minimax and Interlacing Theorems for Matrices. Linear Algebra and Its Applications, 54, 153-172.https://doi.org/10.1016/0024-3795(83)90211-2 [3] Dax, A. (2010) On Extremum Properties of Orthogonal Quotient Matrices. Linear Algebra and Its Applications, 432, 1234-1257 https://doi.org/10.1016/j.laa.2009.10.034 [4] Dax, A. (2019) Bounding Inequalities for Eigenvalues of Principal Submatrices. Advances in Linear Algebra & Matrix Theory, 9, 21-34.https://doi.org/10.4236/alamt.2019.92002 [5] Fan, K. (1949) On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I. Proceedings of the National Academy of Sciences of the United States of America, 35, 652-655.https://doi.org/10.1073/pnas.35.11.652 [6] Fan, K. (1951) Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators. Proceedings of the National Academy of Sciences of the United States of America, 37, 760-766. https://doi.org/10.1073/pnas.37.11.760 [7] Fan, K. (1953) A Minimum Property of the Eigenvalues of a Hermitian Transformation. The American Mathematical Monthly, 60, 48-50.https://doi.org/10.2307/2306486 [8] Horn, R.A. and Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press, Cambridge. [9] Horn, R.A. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.https://doi.org/10.1017/CBO9780511840371 [10] Marshall, A.W., Olkin, I. and Arnold, B.C. (2011) Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics, 2nd Edition, Springer, New York. https://doi.org/10.1007/978-0-387-68276-1 [11] Parlett, B.N. (1980) The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cli s, NJ. [12] Queir o, J.F. (1987) On the Interlacing Property for Singular Values and Eigenvalues. Linear Algebra and Its Applications, 97, 23-28.https://doi.org/10.1016/0024-3795(87)90136-4 [13] Queir o, J.F. and Duarte, A.L. (2009) Imbedding Conditions for Normal Matrices. Linear Algebra and Its Applications, 430, 1806-1811.https://doi.org/10.1016/j.laa.2008.08.015 [14] Thompson, R.C. (1972) Principal Submatrices IX: Interlacing Inequalities for Singular Values of Submatrices. Linear Algebra and Its Applications, 5, 1-12.https://doi.org/10.1016/0024-3795(72)90013-4 [15] Zhang, F. (1999) Matrix Theory: Basic Results and Techniques. Springer-Verlag, New York.