Page 86 - CatalogNEP-PS

 The concept of positive definite matrices and have the basic properties of Positive definite
matrices.

LEARNING OUTCOMES:

On completion of this course the students shall be able to:
 Assimilate the basic idea of operators on finite dimensional vector spaces and the basic
properties of Normal operators in the context of spectral theory.
 Characterize the diagonalizable matrices and have the basic properties of these matrices.
 Acquire the knowledge of the basic concept of matrix norms, their examples, and the
unitarily invariant norm.
 Characterize the positive definite matrices and have the basic properties of Positive
definite matrices. Attain the working knowledge of inequalities involving positive definite
matrices.

THEORY (45 Hours)

UNIT I (15 Hours)

Inner product, Inner product spaces, Linear functional and adjoints, orthogonal projections, self-
adjoint operators. Unitary operators, Normal operators, Spectral theory, functions of operators.
Polar decomposition.

UNIT II (15 Hours)
Simultaneously Diagonalizable Matrices, Unitary equivalence, some implication of Schur‘s
theorem, the eigenvalues of sum and product of commuting matrices. Normal matrices, spectral
theorem for normal matrices, simultaneously unitarily diagonalizable commuting normal matrices.
Matrix norms, Examples, Operator norms, Matrix norms induced by vector norms, The spectral
norm, Frobenius norm, Unitary invariant norm, The maximum column sum matrix norms, the
maximum row sum matrix norm.
UNIT III (15 Hours)

Positive definite matrices, Definitions and properties, Characterizations, The positive semi-
definite ordering, Inequalities for the positive definite matrices, Hadamard‘s inequality, Fischer‘s
inequality, Minkowski‘s inequality.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READING:

1. Linear Algebra, Kenneth Hoffman and Ray Kunze, Second Edition (2001), Princeton-Hall
of India.
2. Matrix Analysis, Roger A. Horn and Charles R. Johnson. Second Edition (2013).
Cambridge University Press.
3. Matrix Analysis, Rajendra Bhatia, Springer Verlag, (1997).
4. Positive Definite Matrix, Rajendra Bhatia, Hindustan Book Agency, (2007).

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