### Mathematical Methods Lecture 6 of 34

September 23, 2011 by K.S. Narain

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.
• ·Degeneracy of eigen vector with rank 1 is given.
• Transformations for the explicit Matrices for making them diagonal is expressed.

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