×

Refinements of the Hadamard and Cauchy-Schwarz inequalities with two inequalities of the principal angles. (English) Zbl 1431.51011

In this paper, the authors explore the relation of the well-known Hadamard and Cauchy-Schwarz inequalities. Based on it, some refinements of the Hadamard and Cauchy-Schwarz inequalities are suggested. Using the volume formulae, a class of principal inequalities related to a parallelotope determined by a matrix is established. This class of principal inequalities have a close relation to the determinant Hadamard and Fischer inequalities. A class of principal inequalities of two subspaces is given using the interlacing property and discussing the principal angles of two subspaces. Using this principal inequality, the Koteljanskii determinant inequality can be proved.

MSC:

51M16 Inequalities and extremum problems in real or complex geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. M. ZHANG, S. G. YANG, A new volume formula for a simplex, Journal of Zhejiang University (Science Edition) 35 (1) (2008) 5-7. · Zbl 1199.52016
[2] G. S. LENG, Y. ZHANG, B. L. MA, Largest parallelotopes contained in simplices, Discrete Mathematics, 211 (2000) 111-123. · Zbl 0955.52006
[3] E. GOVER, N. KRIKORIAN, Determinants and the volumes of parallelotopes and zonotopes, Linear Algebra and its Applications 433 (1) (2010) 28-40. · Zbl 1194.52009
[4] A. BEN-ISRAEL, A volume associated with m×n matrices, Linear Algebra and Its Applications, 164 (1992) 87-111. · Zbl 0762.15003
[5] H. X. LI, G. S. LENG, A matrix inequality with weights and its applications, Linear Algebra and Its Applications, 185 (1993) 273-278. · Zbl 0768.15014
[6] G. S. LENG, Y. ZHANG, The generalized sine theorem and inequalities for simplices, Linear Algebra and Its Applications, 278 (1-3) (1998) 237-247. · Zbl 0937.51013
[7] H. C. YIN, H. M. ZHANG, Properties of bisection planes of dihedral angles of a simplex in the n dimensional Euclidean space, Journal of Zhejiang University (Science Edition) 39 (1) (2012) 18-19.
[8] J. SHENG, Angles between Euclidean subspaces, Geometriae Dedicata 63 (2) (1996) 113-121. · Zbl 0860.51008
[9] B. ZHOU, Z. Y. LI, G. R. DUAN, Y. WANG, Weighted least squares solutions to general coupled Sylvester matrix equations, Journal of Computational and Applied Mathematics 224 (2) (2009) 759- 776. · Zbl 1161.65034
[10] A. G. WU, L. L. LV, G. R. DUAN, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Applied Mathematics and Computation 217 (21) (2011) 8343-8353. · Zbl 1222.65041
[11] H. M. ZHANG, Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications, Computers & Mathematics with Applications 70 (8) (2015) 2049- 2062. · Zbl 1443.65059
[12] ZHAGNH M, DINGF., Iterative algorithms for X+ ATX−1A= I by using the hierarchical identification principle, Journal of the Franklin Institute, 2016, 353 (5): 1132-1146. · Zbl 1336.93066
[13] Z. Y. LI, B. ZHOU, Z. L. LI, On exponential stability of integral delay systems, Automatica 49(11) (2013) 3368-3376. · Zbl 1315.93066
[14] B. ZHOU, W. X. ZHENG, G. R. DUAN, Stability and stabilization of discrete-time periodic linear systems with actuator saturation, Automatica 47(8) (2011) 1813-1820. · Zbl 1226.93107
[15] R. A. HORN, N. H. RHEE, S. WASIN, Eigenvalue inequalities and equalities, Linear Algebra and its Applications 270 (1-3) (1998) 29-44. · Zbl 0890.15019
[16] Q. K. KONG, A. ZETTL, The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory, Linear Algebra and its Applications 434(7) (2011) 1648-1655. · Zbl 1210.15011
[17] H. SCHULZ-BALDES, Sturm intersection theory for periodic Jacobi matrices and linear Hamiltonian systems, Linear Algebra and its Applications 436(3) (2012) 498-515. · Zbl 1232.15010
[18] D. M. ZHOU, G. L. CHEN, G. X. WU, X. Y. ZHANG, On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra and its Applications 438(3) (2013) 1415-1426. · Zbl 1262.15022
[19] X. Z. ZHAN, On some matrix inequalities, Linear Algebra and its Applications 376(1) (2004) 299- 303. · Zbl 1052.15016
[20] J. M. MIAO, A. BEN-ISRAEL, On principal angles between subspaces inRn, Linear Algebra and Its Applications, 171 (1992) 81-98. · Zbl 0779.15003
[21] G. H. GOLUB, C. F. VANLOAN, Matrix Computations, 3rd ed., Baltimore, MD: Johns Hopkins University Press, 1996. · Zbl 0865.65009
[22] X. D. ZHANG, Matrix Analysis and Applications, Beijing: Tsinghua University Press, 2004.
[23] R. BELLMAN, Introduction to Matrix Analysis, Newyork: Mcgraw-Hill Book Company, 1970. · Zbl 0216.06101
[24] C. LUPU, D. SCHWARZ, Another lool at some new Cauchy-Schwarz type inner product inequalities, Applied Mathematics and Computation 231 (2014) 463-477. · Zbl 1410.26041
[25] Z. Z. YAN, A unified version of Cauchy-Schwarz and Wielandt inequalities, Linear Algebra and its Applications 428(8-9) (2008) 2079-2084. · Zbl 1141.15019
[26] R. A. HORN, R. MATHIAS, Cauchy-Schwarz inequalities associated with positive semidefinite matrices, Linear Algebra and its Applications 142 (1990) 63-82. · Zbl 0714.15012
[27] D. R. JOCIC´, S. MILOˇSEVIC´, Refinements of operator Cauchy-Schwarz and Minkowski inequalities for p -modified norms and related norm inequalities, Linear Algebra and its Applications 488 (2016) 284-301. · Zbl 1343.47012
[28] S. H. WADA, On some refinement of the Cauchy-Schwarz inequality, Linear Algebra and its Applications 420 (2007) 433-440. · Zbl 1121.47010
[29] J. M. ALDAZ, S. BARZA, M. FUJII, M. S. MOSLEHIAN, Advances in operator Cauchy-Schwarz inequalities and their reverses, Annals of Functional Analysis, 6 (3) (2015) 275-295. · Zbl 1312.47022
[30] H. M. ZHANG, F. DING, A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations, Journal of the Franklin Institute 351 (1) (2014) 340-357. · Zbl 1293.15006
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.