### Mathematical Methods Lecture 7 of 34

September 26, 2011 by K.S. Narain

TEST

K.S. Narain , ICTP

lecture 7

Characteristic Polynomials,

• Note that in order to be more specific for this semester several things in chapter one have not been mentioned just like tensor calculation and so on.
• on the other hand some of them was given in tutorial .
• The definition of characteristic polynomials by taking a n*n matrix and imposing nth degree polynomial in $\lambda$ are given.Some examples are also given.
• Defining some subspaces with some special examples have been said.
• Decomposing several vectors as a linear combination of the matrices realized.
• Making some subspaces which consist of generalized eigen vectors with rank less than r used.
• Taking basis for any vector by taking into account generalized eigen vectors has been used.
• We proved that generalized igen vectors from different subspaces are linearly independent(was given in the class).
• We present two examples on how to define degeneracy on a 3*3 Matrix.
• We talked a little bit about hamiltonian theorem.

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