Alg1231W8T2 - Eigenvectors and Eigenvalues of a reflection

Wednesday, 12 September 2018

9:35 PM

Machine generated alternative text: -2 6 4 2 o -2 2 4 x 6 Let's recall that a linear transformation T has an eigenvector v O precisely when T(v) = Av for some scalar which is called its eigenvalue_ We have seen that is the reflection in the line y y and •\'2 ¯ x _ This has eigenvalues Al — 1 1 O with respective eigenvectors and v 2 Recall: the Maple notation for the vector is <1 , 2 We say that Al and are the eigenvalues of the matrix A Let's note that the sum of the eigenvalues is Al + — o 1 1 O and the product of the eigenvalues is — O These two numbers, known as the trace and the determinant of A, play an interesting role in transformations of the plane.

 

Machine generated alternative text: 2 -2 R(v) 14 4 Let R be the linear map which reflects in R The other eigenvalue is — The matrix that represents T is The characteristic polynomial for B is Recall: 1 • the Maple notation for the matrix 3 1 • the Maple notation for the vector 2 about the line y —x. Geometrically we can see that an eigenvector of R associated to the eigenvalue Al — 1 is the vector O with associated eigenvector = det(B - ti) 2 4 is

 

Machine generated alternative text: 2 o -4 --2 -2 S(v) 2 x 4 —4 Consider the transformation S : IR2 R which is reflection in the origin Then S has matrix representation S(v) cv where The characteristic polynomial of C is One could say this has zeros Al — multiplicity 2 p(t) = det(C — tI) = O but in such a case we prefer to say that C has a single eigenvalue O with In this case, an associated eigenvector for C is (pick all that apply) 2 1 -2 1 1 1 2 any non-zero vector in R 2 Cl any vector in IR 1 Recall: the Maple notation for the matrix 3 2 112 >,< 314 4

 

Machine generated alternative text: Consider the vector space PI = {ax + b : a, b e R} , i.e. the space of polynomials of degree at most 1. Let T : PI PI be the map T(ax + b) :— Then T(4x + 7) — T(-3C+4) — 4X-3 If we identify these linear polynomials with vectors via and then T(ax + b) ax + b b and hence T has matrix representation o b 1 This matrix has characteristic polynomial p(t) = det(A — tI) From this we can determine that the linear transformation T has the set of eigenvalues eigenvectors {X+l ,x-l} Recall: • to enter a set like {1, 2} in Maple notation use the syntax {1, 2} • multiplication is denoted using an asterisk * 1 t"2-1 {1,-1} O and, expressed in polynomial form, an associated set of • the vectors in the vector space PI are polynomials, so the vector 2x + 1 would be entered in the form 2*x+1.

 

Machine generated alternative text: Consider the 4-dimensional vector space of 2 x 2 matrices a M2,2 (R) - b : a,b,c,deR d Define the transformation You may remember this matrix transformation is called the transpose: T(A) c AT Now b d It follows that T has 3 eigenvalue —1. O linearly independent eigenvectors associated to the eigenvalue 1 and 1 O linearly independent eigenvectors associated to the 1 Recall: the Maple notation for the matrix 3 2 112 >,< 314 4

 

 

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