K.S. Narain , ICTP

### lecture 6

*Proving orthonormality using hermition operators, Map beween two subspace, Transformations for Matrices*

- 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 $\lambda_i$ . Mentioned how many null
- vectors must be existed in subspaces.
*ยท**Degeneracy of eigen vector with rank 1 is given.**Transformations for the explicit Matrices for making them diagonal is expressed.*