Abstract
Extremal representations for the maximum and minimum eigenvalues of a symmetric matrix are proved. Singular values are defined and the Singular Value Decomposition is obtained. Courant–Fischer Minimax Theorem, Cauchy Interlacing Principle and majorization of diagonal elements by eigenvalues of a symmetric matrix are proved. The volume of a matrix is defined as the positive square root of the product of the nonzero singular values. Some basic properties of volume are proved. Minimality properties of the Moore–Penrose inverse involving singular values are established.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Bapat, R.B. (2012). Inequalities for Eigenvalues and Singular Values. In: Linear Algebra and Linear Models. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2739-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2739-0_5
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2738-3
Online ISBN: 978-1-4471-2739-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)