**DESCRIPTION OF COURSES**

**AS 551 MATHEMATICAL METHODS IN STATISTICS (4L+0P)**

**Objective**

This is another course that supports all other courses in Statistics / Agricultural Statistics. The students would be exposed to the advances in linear algebra and matrix theory. This would prepare them to study their main courses that involve knowledge of linear algebra and matrix algebra.

**Theory**

UNIT I

Linear Algebra: Group, ring, field and vector spaces, sub-spaces, basis, Gram Schmidt’s Orthogonalization, Galois field - Fermat’s theorem and primitive elements. Linear transformations.

UNIT II

Real Analysis: Convergence and divergence of infinite series, use of comparison tests - D’Alembert’s Ratio - test, Cauchy’s nth root test, Raabe’s test, Kummer’s test, Gauss test. Absolute and conditional convergence. Riemann integration, concept of Lebesgue integration, power series, Fourier, Laplace and Laplace -Steiltjes’ transformation, multiple integrals.

UNIT III

Matrix Algebra: unitary, similar, hadamard, circulant, helmert’s matrices. Kronecker and hadamard product of matrices. Sub-matrices and partitioned matrices, permutation matrices, full rank factorization. Equations having many solutions.

UNIT IV

Generalized inverses, Moore-Penrose inverse, applications of g-inverse. Spectral decomposition of matrices, Differentiation and integration of matrices, Quadratic forms. Graph theory: Concepts and applications. Fuzzy set theory.

**Suggested Readings**

- Aschbacher, M. 2000.
*Finite Group Theory*. Cambridge University Press. - Deo, N. 1984.
*Graph Theory with Application to Engineering and Computer Science*. Prentice Hall of India Pvt. Ltd. - Frank, A. 1962.
*Schaum’s Outline of Theory and Problems of Matrices*. McGraw-Hill. - Frank, H. 1994.
*Graph Theory*. Addison Wesley. - Gentle, J. E. 2007.
*Matrix Algebra: Theory, Computations and Applications in Statistics*. - Springer. Graybill, F.E.1961.
*Introduction to Matrices with Applications in Statistics*. - Wadsworth Pub. Co. Hadley, G. 1969.
*Linear Algebra*. Addison Wesley. - Harville, D.A. 1997.
*Matrix Algebra from a Statistician’s Perspective*. - Springer. John B. F. 2002.
*A First Course in Abstract Algebra*. Addison Wesley. - Khinchin, A. I.1957.
*A Course of Mathematical Analysis*. Gordon & Breach Science Pub. - Malik, S.C. and Arora, S. 1992.
*Mathematical Analysis*. New Age International. - Narayan, S. 1953.
*A Text Book of Matrices.*S. Chand and Company. - Rao, C.R. 1965.
*Linear Statistical Inference and its Applications*. John Wiley. - Robinson, D.J.S. 1991.
*A Course in Linear Algebra with Applications*. World Scientific, Singapore. - Royden, H.L. 1968.
*Real Analysis*. Macmillan Library Reference. - Searle, S.R. 1982.
*Matrix Algebra Useful for Statistics*. John Wiley. - Seber, G.A.F. 2000.
*A Matrix Handbook for Statisticians*. John Wiley. - Thomson, B.S., Bruckner, J.B., Bruckner A.M. 2007.
*Elementary Real Analysis*. Jones and Bartlett. - Wilson, R. J. 1972.
*Introduction to Graph Theory*. Academic Press.