Linear Algebra - II

Code: MA719 | L-T-P-C: 4-0-0-8

MA719 Linear Algebra - II [4-0-0-8] Prerequisites: Nil

Eigenvalues and eigenvectors, Schur's theorem - real and complex versions, Spectral theorems for normal and Hermitian matrices - real and complex versions. Gerschgorin discs with associated perturbation theorems and inclusion results. Jordan canonical forms with application, minimal polynomials, companion matrices. Functions of matrices via spectral decompositions. Variational characterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischer theorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law of inertia. Matrix norms, spectral radius formula, relationships between matrix norms. Singular value decomposition, polar decomposition. Positive definite matrices, characterizations of definiteness, polar form and singular value decompositions, congruence and simultaneous diagonalization.

Texts:

1. R. A. Horn and C. R. Johnson, Matrix Analysis, CUP, 1985.
2. S. Axler, Linear Algebra Done Right, 2nd Edition, UTM, Springer, Indian Edition, 2010.
3. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second edition, Academic Press, 1985.
4. F. R. Gantmacher, The Theory of Matrices, Vol-I, Chelsea, 1959.