Mathematics 1600A/B Lecture Notes - Lecture 28: Laplace Expansion, Polynomial, Diagonalizable Matrix

• School: Western University
• Department: Mathematics
• Course: Mathematics 1600A/B
• Professor: Graham Denham

Document Summary

Similarity and diagnoalisation: compute the characteristic polynomial p( ) If you just calculate this polynomial as a machine, you will have a degree 4 polynomial but you need to find its roots: you could use cofactor expansion p( ) = ( - 2)2( -3)( +1) Algebraic multiplicity of 2: 2: - 22 divides the polynomial. Geometric multiplicity: the dimension of the corresponding eigenspace: = -1, = 2, = 3. If you are given a matrix, that matrix could be complicated. But if you choose your coordinates well (from a basis of eigenvectors), that matrix becomes very simple. Similarity definition: two n x n matrices are said to be similar if there is an invertible n x n matrix p so that p-1ap = b we write a ~ b. We should think of p as a change of coordinates. A and b are the same up to a change of coordinates.