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Sophie Chrysostomou

University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT B24S Fall 2011 Lecture 9 Suppose that T : V −→ V is a linear transformation from vector space V with ordered basis B into the vector space V with ordered basis B . Let ′ ′ [T] B, B′ be the matrix representation of T relative to B amd B . Now let ′ ′ ′ T : V −→ W be a linear transformation from the vector space V to the vector space W with ordered basis B . Then [T ◦T] ′ ′′= [T ]′ ′ ′[T] ′ B,B B , B B, B This follows from the fact that for all v ∈ V and v ∈ V we have that 1. (T(v)) B′ = [T] B,B ′vB ′ ′ ′ ′ 2. (T (v )) B” = [T ] B ,B′v B′ ′ 3. let v = T(v) ▯ ▯ ′ ′ ′ ′ 4. (T ◦T)(v) B” = (T (T(v))) B” = [T ] B B” (T(v)) B′ = [T ] B ,B′′[T]B,B ′vB ′ Now suppose that B and B are two ordered bases of V and T : V −→ V is a linear transformation. ′ Suppose that C B, B ′ is the change of coordinate matrix from B to B . Then for all v ∈ V we have v ′ = C ′v . Suppose also that C ′ is B B, B B B , B the change of coordinate matrix from B to B. Then for all v ∈ V we have v B C B , Bv B . If [T] B is the matrix rep of T relative to B only i.e. T(v) B = [T] vB B ′ for all v ∈ V. and that [T] B ′ is the matrix rep of T relative to B only i.e. T(v) B′ = [T] B ′vB ′ for all v ∈ V. Then we have that −1 [T]B= C B, B′[T]B ′C B, B′ Thus [T] ′ and [T] are similar matrices. B B 1 Remember that D is a similar matrix to A if D can be obtained from A by performing elementary row (or column) operations. Now we see that they are also both matrix rep of the same linear transformation T relative to suitable different ordered basis. In the same way then it may be possible to find the matrix rep of T rel to some ordered basis such that this matrix is a diagonal. Remember also that similar matrices have the same eigenvalues. (ie. λ is an eigenvalue of A if Av = λv for some eigenvector v of A) We say that a lin trans T has an eigenvalue λ if T(v) = λv and we call v an eigenvector of T. Review Chapter 5 from last year. It is sometimes of interest and convenient to have the matrix rep of a linear transformation relative to an ordered basis B in a diagonal form. This is only possible if a matrix rep of the linear transformation relative to a known basis is diagonalizable. Suppose we have A the matrix rep of T relative to some known basis ′ B . We need to find if it is possible to find a diagonal matrix which is the representation of T relative to some basis B we need to find out if A is diagonalizable. Thus we need to do the following: 1. We find the characteristic polynomial |A − λI|. 2. We find the eigevalues of A which are the roots of the characteristic polynomial of A. 3. Find the algebraic multiplicity of each of the eigenvalues. 4. For each eigenvalue λ fini the eigenspace E λiwhich is the nullspace of A − λ .i 5. The dim(E ) iλithe geometric multiplicity of λ . i 6. If the geometric multiplicity of λ = tie algebraic multiplicity of λ for i each λ ,ithen A is diagonalizable! 2 Once we determine that A is diagonalizable, then we construct C the matrix whose columns are the bases vectors of each eigenspace E . Theλi −1 D = C AC is a diagonal matrix. The jth column of D has a nonzero entry the eigenvalue whose eigenvector is the column j of the matrix C. We have also established that A = [T] B′ and that D is a representation of T relative to some basis B as well. In other words: D = C −1 [T] C ′ B, B′ B B, B Thus C =
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