Download MAT-653

MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-653
(3L+1T+0P)
Advanced Matrix Theory
4 Credits
Quadratic forms and congruence of Matrices:- Quadratic forms, Quadratic forms as a product
of matrices, Matrices as representative of linear transformation, the set of quadratic forms
over F, congruence of quadratic forms and matrices. Congruence transformation of
symmetric matrix. Elementary congruent transformations, congruent reduction of a
symmetric matrix, congruence of skew symmetric matrices.
Quadratic forms in the real field:- Reduction in the real field, classification of real quadratic
forms in n-variables, definite, semi definite and indefinite real quadratic forms. Quadratic
characteristics properties of definite, semi definite forms, gram matrices, case of complex
field, reduction in the complex field.
Hermitian matrices and forms:- Hermitian matrices and forms, linear transformation of
Hermitian form, conjunctive transformation of a matrix, conjunctive reduction of Hermitian
matrix, types of Hermitian forms, conjunctive reduction of a Hermitian matrices.
Characteristic roots and characteristic vectors of matrices:- Characteristic roots and
characteristic vectors of a square matrix, Nature of the characteristic roots of special types of
matrices, relation between algebraic and geometric multiplicities of characteristic roots,
mutual relation between characteristic vectors corresponding to different characteristic roots.
Books recommended:
1. S. Barnett ,Matrix Methods for Engineers and Scientists,McGraw-Hill,London,(1979).
2. Eves, Howard ,Elementary Matrix Theory,Dover Publication,(1980).
3. Shanti Narayan and P.K.Mittal,Text book on Matrix Theory,S.Chand,(1953).