Activity 8: Eigenvectors and Linear Transformations MATH 20, Fall 2010 November 1, 2010 Names: In this activity, we will explore the effects of matrix diagonalization on the associated linear transformation. We start by considering a diagonal matrix and the coordinate system associated to the standard basis of n. 1. Let A 3 0 0 2 and let x x y be an arbitrary vector in 2. a) What is the standard basis for 2? b) How would you write x as a linear combination of the standard basis vectors of 2? c) Now write A x as a linear combination of the standard basis vectors of 2. Notice that multiplication by A is the same as scaling the e 1 -component of x by 3 and the e 2 -component by 2. d) Let k be a positive integer. How would you write A k x as a linear combination of the standard basis vectors of 2? 2 v 1, v 2 v 1 1 1 v 2 1 2 v 1 v 2 2
2 Eigenvectors and linear transformations.nb u 1 u u 4 v 1 1 v 2 w w v 1 v 2 x x y x a b x 4 v 1 1 v 2 p p v 1 v 2 v 1 x 4 v 4 0 0 1 For problems 3-, let A 3 1 2 2, and let T : 2 2 be the linear transformation given by Tx A x. 3. a) Find the eigenvalues Λ 1 and Λ 2 of A. Assume Λ 1 Λ 2. b) Find a basis for the eigenspace of A associated to Λ 1. c) Find a basis for the eigenspace of A associated to Λ 2. Clear denominators if needed so that your basis vector has integer entries. d) Do the vectors found in parts b) and c) form a basis for 2? 4. a) Diagonalize A. That is, find an invertible matrix P and a diagonal matrix D so that A P D P 1. P ; Diag ; b) Find P 1. Pinv ;
Eigenvectors and linear transformations.nb 3 c) Letting b 1, b 2 be the eigenvector basis for 2 that you found in parts b) and c), find Tb 1 and Tb 2. d) What are Tb 1 and Tb 2? e) How does your answer to part d) relate to the diagonal matrix you found in part a)? We call D the -matrix for the transformation Tx A x, because in the -coordinate system with basis vectors b 1 and b 2, the transformation T can be written as multiplication by D. The notation for this is T D, or Tx Tb 1 Tb 2 x Dx.. Let P be the matrix you found in 4.a). P 1 1? a) Write 1 as a linear combination of b 1 and b 2 (Compare to problem 2.b)). What is b) Notice that an arbitrary vector x could be written as Px x., where P is the matrix you found in 4.a). What is P 1 x? c) Using part b), 4.e), and the notation above problem., write Tx in terms of D and P 1. d) Simplify PTx. (See part b) of this problem.) Putting this all together, we see that diagonalizing a matrix A has the following effect: i) Multiplication by P 1 takes the vector x into the eigenvector coordinate system. ii) Multiplication by D then scales the component in the b i -direction by the eigenvalue Λ i. iii) Multiplication by P finally sends the vector back to the original coordinate system: the standard basis of n. Written in symbols, A x P D P 1 x P Dx PTx Tx.
4 Eigenvectors and linear transformations.nb 6. Consider the matrix A 12 2 3 13. The eigenvalues of A are 2 and 3, with respective in the - eigenvectors b 1 3 2 and b 2 1 1. The diagram below depicts the vector v 2 3 coordinate system, where 3 2, 1 1. P 3 1 ; ShowPlot2, x, 1, 10, PlotRange 1, 10, AspectRatio Automatic, GraphicsTableGray, Dashed, LineP.0 k, 10, P.0 k, 20, k, 20, 20, TableGray, Dashed, LineP.10, 0 k, P.20, 0 k, k, 20, 20, GraphicsBlue, Thick, Arrow0, 0, 2, 3 8 6 4 2 2 4 6 8 10
Eigenvectors and linear transformations.nb a) What are the -coordinates of v 2 3? 2 3 b) Using the fact that the eigenvalues of A are 2 and 3, respectively, what are the coordinates of A 2 v? (Remember that multiplication by A means scaling the b 1 -component of the vector by 2 and the b 2 -component by 3.) A 2 v 7. a) Find A 2 v. (Recall that Px x, where P 3 1 A 2 v, the matrix of eigenvectors.) b) Check to see that your answer makes sense by graphing it as an arrow in the coordinate system below: (Fill in the standard basis coordinates for the missing red vector.) P 3 1 ; Show Plot2, x, 1, 20, PlotRange 1, 20, AspectRatio Automatic, GraphicsTableGray, Dashed, LineP.0 k, 10, P.0 k, 20, k, 20, 20, TableGray, Dashed, LineP.10, 0 k, P.20, 0 k, k, 20, 20, Graphics Red, Thick, Dashed, Line0, 0, P.2^2, 0, P.2^2, 3^2, Blue, Thick, Arrow0, 0, 2, 3, Red, Thick, Arrow0, 0,, Recap of problems 6 and 7: i) In 6.a), we applied P 1 to v. ii) In 6.b), we applied the diagonal matrix D 2 to the previous result. iii) In 7.a), we applied P to that result. iv) Overall effect: P D 2 P 1 v A 2 v. 8. Check your results by using Mathematica to compute A 2 v, with A and v as in problem 6.