OSU MTH 254 - Eigenvalues, Eigenvectors and Quadratic Forms

Unformatted text preview:

Eigenvalues, Eigenvectors and Quadratic Forms: ExamplesMth 254 Mar 12 2001Bent E. PetersenFilename: 254w2001_eigenvects_examp.mwsLinear Algebra supplement for Vector Calculus 1.In this worksheet I give a few examples of eigenvalues and eigenvectors of matrices in the first part. In the second part I provide the answeres, as given by Maple, to the problems in the Mth 254 Study Guide (Parks 2000-01 revision).If you try a few of these examples by hand you will develop a deep appreciation for Maple, or a similar tool. > restart;> with(linalg):Warning, new definition for normWarning, new definition for tracePart 1. ExamplesExample 1> A1:=matrix(3,3,[-1,2,2,2,2,2,-3,-6,-6]); := A1-1 2 22 2 2-3 -6 -6> eigenvals(A1);, ,0 -3 -2A1 has distinct eigenvalues. Thus A1 is diagonizable. Let's diagonalize it explicitly.> v1:=eigenvects(A1); := v1 , ,[ ], ,-3 1 { }[ ], ,-1 0 1 [ ], ,0 1 { }[ ], ,0 -1 1 [ ], ,-2 1 { }[ ], ,-2 1 0> s1:={}: for k from 1 to 3 do s1:=s1 union op(3,op(k,[v1])): od: unassign('k'): s1;{ }, ,[ ], ,-1 0 1 [ ], ,0 -1 1 [ ], ,-2 1 0> S1:=matrix(3,0,[]); := S1 ( )array , , .. 1 3 .. 1 0 [ ]> S1:=augment(S1,op(s1));Page 1:= S10 -1 -2-1 0 11 1 0> inverse(S1) &* A1 &* S1: D1:=evalm(%); := D10 0 00 -3 00 0 -2Alternate Calculation for Example 1There is an easier way to do these calculations. Let's start over. First we compute the eigenvaules of A1 and put them in a diagonal matrix. We can do this as follows:> D1b:=diag(eigenvals(A1)); := D1b0 0 00 -3 00 0 -2Now if A1 is diagonizable the it will be 'similar' to D1b - so we ask Maple to check:> issimilar(A1,D1b,'P1b');trueThe return value indicates A1 is diagonizable. In this case P1b has been assigned a matrix which plays the role of the inverse of S1 above. let's check it;> evalm(P1b); S1b:=inverse(P1b);161316-13-23-23-1 -1 -1 := S1b0 3 -26 0 1-6 -3 0It doesn't look the same, but let's check that it works:> inverse(S1b) &* A1 &* S1b: evalm(%);Page 20 0 00 -3 00 0 -2That worked as expected.Example 2> A2:=matrix(3,3,[7,4,-4,4,-8,-1,-4,-1,-8]); := A27 4 -44 -8 -1-4 -1 -8> eigenvals(A2);, ,9 -9 -9This time we have an eigenvalue (-9) with algebraic multiplicity 2. We do not yet know if A2 is diagonizable or not. Let's compute the eigenvectors.> v2:=eigenvects(A2); := v2 ,[ ], ,9 1 { }[ ], ,4 1 -1 [ ], ,-9 2 { },[ ], ,0 1 1 [ ], ,1 -4 0We see that -9 has geometric multiplicity 2. Thus A is diagonizable. Let's diagonalize it explicitly.> s2:=op(3,op(1,[v2])) union op(3,op(2,[v2])); := s2 { }, ,[ ], ,0 1 1 [ ], ,1 -4 0 [ ], ,4 1 -1> S2:=matrix(3,0,[]); := S2 ( )array , , .. 1 3 .. 1 0 [ ]> S2:=augment(S2,op(s2)); := S24 1 01 -4 1-1 0 1> inverse(S2) &* A2 &* S2: D2:=evalm(%); := D29 0 00 -9 00 0 -9Again, as expected.Example 3> A3:=matrix(3,3,[1,1,-1,-1,3,-1,-1,2,0]);Page 3:= A31 1 -1-1 3 -1-1 2 0> eigenvals(A3);, ,2 1 1Again we have an eigenvalue (1) with algebraic multiplicity 2. We do not yet know if A3 is diagonizable or not. Let's compute the eigenvectors.> s3:=eigenvects(A3); := s3 ,[ ], ,1 2 { }[ ], ,1 1 1 [ ], ,2 1 { }[ ], ,0 1 1Oops. We see the eigenvalue 1 has geometric multiplicity 1 (only one eigenvector for 1 is listed by Maple). Thus A3 is not diagonizable. The "closest" we can get it the Jordan canonical form> jordan(A3);2 0 00 1 10 0 1You probably suspect we could have used the jordan() function above to verify the diagonizability of A1 and A2. If so, you are correct, but it would have been less fun.> jordan(A1);-2 0 00 -3 00 0 0> jordan(A2);-9 0 00 9 00 0 -9Example 4This time consider a symmetric matrix. We know that a symmetric matrix has real eigenvalues and is diagonizable whether it has repeated eigenvalues or not.> A4:=matrix(4,4,[0,1,-1,2,1,1,0,-1,-1,0,-1,1,2,-1,1,0]);Page 4:= A40 1 -1 21 1 0 -1-1 0 -1 12 -1 1 0If we compute the exact eigenvalues we get a nearly incomprehensible expression, so let's compute approximate eigenvalues instead.> evalf(eigenvals(A4),16);, , ,2. 1.681330643604977 − − 3.323404276086478 .2 10-15I − − .357926367518500 .2 10-15IThose imaginary parts should not be there! They are due to roundoff as you can see by doing the calculation at several precisions. For example> evalf(eigenvals(A4),22);2. − 1.681330643604977375475 .1 10-21I − − 3.323404276086477625772 .1 10-21I, , ,− − .357926367518499749704 .1 10-21IThat A4 is diagonizable we can verify by computing the Jordan canonical form> evalf(jordan(A4));2. 0 0 00 − 1.681330643 .3 10-9I 0 00 0 -3.323404277 00 0 0 -.357926367Example 5Here is another symmetric example:> A5:=matrix(4,4,[0,1,2,3,1,0,1,2,2,1,0,1,3,2,1,0]); := A50 1 2 31 0 1 22 1 0 13 2 1 0> jordan(A5);Page 5− + 2 2 0 0 00 − − 2 2 0 00 0 + 2 10 00 0 0 − 2 10We see A5 is diagonizable as expected.Part 2. Solutions for the Study Guide, Lesson 22Here are Maple's solutions for the problems in Lesson 22. The matrices are numbered in accord with the problem numbers. You may note some trivial differences between the solutions and the selected solutions in the Study Guide. Keep in mind any nonzero multiple of an eigenvector is an eigenvector. More generally, any nonzero linear combination of eigenvectors corresponding to the same eigenvalue, is an eigenvector.After studying the examples above you should be able to decipher Maple's output. Explicitly the eigenvects() function returns a list consisting of ordered triples of the form[eigenvalue, algebraic multiplicity, {linearly independent set of eigenvectors}]The number of vectors in the returned set of eigenvectors is the geometric multiplicity of the eigenvalue. The matrix is diagonizable if and only if the algebraic and geometric multiplicities are equal for each eigenvalue.> M01:=matrix(2,2,[6,2,1,5]); := M016 21 5> eigenvects(M01);,[ ], ,4 1 { }[ ],-1


View Full Document

OSU MTH 254 - Eigenvalues, Eigenvectors and Quadratic Forms

Documents in this Course
Load more
Download Eigenvalues, Eigenvectors and Quadratic Forms
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Eigenvalues, Eigenvectors and Quadratic Forms and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Eigenvalues, Eigenvectors and Quadratic Forms 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?