Mathematics 1600A/B Lecture Notes - Lecture 28: Laplace Expansion, Polynomial, Diagonalizable Matrix
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19 Nov 2018
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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.
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(0, 0) (0, 1) (1, 1) |
121p^2 + 88p - 16 121p^2 - 16 121p^2 - 88p -16 |
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x^2 + 8xy + 8y^2 x^2 + 8xy + 15y^2 x + 8xy + 15y |
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Trinomial, degree 3 Trinomial, degree18 Trinomial, degree11 |
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False |
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False |
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