### Mathematical Methods Lecture 13 of 34

October 13, 2011 by K.S. Narain

K.S. Narain , ICTP

Review of the rest of the lectures will be given but for the direct course see the attached addresses. Some of the contents in this lecture are not in the book. The inverse of Matrix, Eigen value equation, Standard matrix multiplication ,Transformation for covariant and contra variant objects have been said. Determinant of 2*2 and 3*3 Matrix introduced. Writing determinant of a matrix in a compact form in terms of Levi Chevita tensor and its properties and various examples are given. Determinant of product of two matrices is also explained. Showing n-linear inhomogeneous equation for n-unknown quantities and the prove is given by contradiction. By introducing several examples for adjoint , hermition and unitary operators we learn how to come over those things. The definition of scalar is given .we proved that determinant of matrix will not change under changing of the basis . We also prove that the scalars which are invariant just like the trace of a matrix and we discussed cyclic property of the trace. Proving orthonormality using hermition operators, Map beween two subspace, Transformations for Matrices were applied. We discuss any vector can be written in terms of linear combination of eigen vectors. Eigen vectors can form basis so hermitian operator can be diagonalized . We proved that statement which was so long and gave some examples. Characteristic Polynomials are given . Decomposition of full vector space in terms of subspace is given. We proved subspace (S_i) consist of generalized eigen vectors of $A_i$ with eigen value of $\lambda_i$ and mentioned how many null vectors must be existed in subspaces.Degeneracy of eigen vector with rank 1 is given. Transformations for the explicit Matrices for making them diagonal is expressed.

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