A Bauer-Hausdorff matrix inequality. (English) Zbl 1449.15055

Yan, Zizong; Wu, Shanhe

Abstr. Appl. Anal. 2013, Article ID 894824, 4 p. (2013).

Summary: We present a biorthogonal process for two subspaces of \(\mathbb{C}^n\). Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality.

MSC:

15A45	Miscellaneous inequalities involving matrices
15B57	Hermitian, skew-Hermitian, and related matrices

References:

[1]	Eijkhout, V.; Vassilevski, P., The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods, SIAM Review, 33, 3, 405-419 (1991) · Zbl 0737.65026 · doi:10.1137/1033098
[2]	Wielandt, H., Inclusion theorems for eigenvalues, Simultaneous Linear Equations and the Determination of Eigenvalues. Simultaneous Linear Equations and the Determination of Eigenvalues, National Bureau of Standards Applied Mathematics Series, 75-78 (1953), Washington, DC, USA: U.S. Government Printing Office, Washington, DC, USA · Zbl 0052.25602
[3]	Bauer, F. L.; Householder, A. S., Some inequalities involving the euclidean condition of a matrix, Numerische Mathematik, 2, 308-311 (1960) · Zbl 0104.34502 · doi:10.1007/BF01386231
[4]	Wang, S. G.; Ip, W. C., A matrix version of the Wielandt inequality and its applications to statistics, Linear Algebra and Its Applications, 296, 1-3, 171-181 (1999) · Zbl 0942.15014 · doi:10.1016/S0024-3795(99)00117-2
[5]	Auzinger, W.; Kirlinger, G., Kreiss resolvent conditions and strengthened Cauchy-Schwarz inequalities, Applied Numerical Mathematics, 18, 1-3, 57-67 (1995) · Zbl 0834.15020 · doi:10.1016/0168-9274(95)00043-T
[6]	Kreiss, H. O., Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT Numerical Mathematics, 2, 153-181 (1962) · Zbl 0109.34702
[7]	Drury, S. W.; Liu, S.; Lu, C. Y.; Puntanen, S.; Styan, G. P. H., Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments, Sankhyā, 64, 2, 453-507 (2002) · Zbl 1192.15007
[8]	Eaton, M. L., A maximization problem and its application to canonical correlation, Journal of Multivariate Analysis, 6, 3, 422-425 (1976) · Zbl 0332.15008 · doi:10.1016/0047-259X(76)90050-6
[9]	Kantorovič, L. V., Functional analysis and applied mathematics, Uspekhi Matematicheskikh Nauk, 3, 6, 89-185 (1948) · Zbl 0034.21203
[10]	Zhang, F., Equivalence of the Wielandt inequality and the Kantorovich inequality, Linear and Multilinear Algebra, 48, 3, 275-279 (2001) · Zbl 0995.15011 · doi:10.1080/03081080108818673
[11]	Zhan, X. Z., Matrix Theory (2008 (Chinese)), Beijing, China: Higher Education Press, Beijing, China · Zbl 1181.15001
[12]	Fujii, M., Wielandt Theorem: Simple Proof and Its Generalizations. Wielandt Theorem: Simple Proof and Its Generalizations, Hokkaido University Technical Report Series in Mathematics (1997)
[13]	Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0576.15001
[14]	Zhang, F., The Schur Complement and Its Applications. The Schur Complement and Its Applications, Numerical Methods and Algorithms, 4 (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1075.15002 · doi:10.1007/b105056