Eigenvalues and eigenvectors

Transcription

1 Eigenvalue and eigenvector Defining and computing uggeted problem olution For each matri give below, find eigenvalue and eigenvector. Give a bai and the dimenion of the eigenpace for each eigenvalue. P: Solve for eigenvalue: A AI det( AI) (5 )( ) (4)() "characteritic polynomial" ( 6) ( ) det( AI) when 6,. Thee are the eigenvalue of A. Get eigenvector for each eigenvalue in turn by etting up the ytem ( A I ) : : rref Let and o.

2 , i an eigenvector of A, aociated with. All calar multiple of it are alo eigenvector. The et of all eigenvector, called the eigenpace of A aociated with, i the pace generated by,. A bai for thi pace i {, }, and ha dimenion. 6 : rref 4 4 Let and o , i an eigenvector of A, aociated with 6. All calar multiple of it are alo eigenvector. The et of all eigenvector, called the eigenpace of A aociated with 6, i the pace generated by 4,. A bai for thi pace i { 4, }, and ha dimenion. You can get eigenvalue and eigenvector in SciLab by uing the pec command: -->A=[5 4; ]; -->[evec,eval]=pec(a) eval = 6.. evec = The diagonal entrie of the eval matri are the eigenvalue; the column vector of the evec matri are the vector (o.97,.4 i aociated with 6, and.77,.77 with. SciLab normalize it eigenvector into unit vector normalize, for eample, and you ll get,.77,.77.

3 P: A 4 Solve for eigenvalue: AI 4 det( AI) ( )(4 ) ( )() "characteritic polynomial" ( ) ( ) det( AI) when,. Thee are the eigenvalue of A. Get eigenvector for each eigenvalue in turn by etting up the ytem ( A I ) : : 4 rref Let and o., i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,. A bai for thi pace i {, }, and ha dimenion.

4 : 4 rref Let and o., i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,. A bai for thi pace i {, }, and ha dimenion. P: A Solve for eigenvalue: AI det( AI) ( )( ) ()() 4 4 "characteritic polynomial" Thi one i in here to make the point that there no particular reaon to epect thee to be factorable, and that eigenvalue can be irrational. Ue the quadratic formula: And the eigenvalue of A are and.

5 Be careful when olving for eigenvector it natural to want to go to decimal and round, but look at what happen if you do that: Try.68 : rref??.7?? According to SciLab, that coefficient matri reduce to identity -->A=[.7 ;.7 ]; -->rref(a) an = Which would make the olution, and give an eigenvector of,. But we know that thi cannot be correct; by definition, eigenvector are nonzero, and the way we derived the whole proce from the outet wa by requiring that det( A I), meaning that A I i a ingular matri and cannot reduce to identity. The homogeneou ytem mut have infinitely many olution. There nothing wrong with SciLab; the matri I entered in i in fact noningular. The point i that i not the ame number a.7, and rounding on eigenvector problem i enough to completely me up the olution. Stay eact : ( ) ( ) rref

6 It worth noting that Scilab ha enough preciion that if you enter the quare root a quare root intead of pre rounding, it ll rref correctly. -->A=[-+qrt() ; +qrt() ]; -->rref(a) an = However, you hould alo note that I wa able to rref thi rref immediately without performing any algebra at all (and you hould be able to a well). Since we know up front that the coefficient matri mut be ingular, and thi i a ytem, it mut be the cae alway that the row are calar multiple of each other. In an eigenvalue problem, you re guaranteed a bottom row of zero, and the urviving top row can be either of the initial two row, or any calar multiple of them. Since the econd row already had a leading, I knew that in rref d form all that would happen would be that it would move to the top, and the bottom row would zero out. Anyway, back to the eigenvector Let and o ( ). ( ) ( ) ( ), i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by ( ),. A bai for thi pace i { ( ), }, and ha dimenion.

7 : ( ) ( ) rref Let and o ( ). ( ) ( ) ( ), i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by ( ),. A bai for thi pace i { ( ), }, and ha dimenion. P4: A Solve for eigenvalue: AI det( AI) ( ) I m uing cofactor epanion acro the top row to get the determinant. Thi may not alway be the bet approach, ince we re going to end up with a cubic polynomial. How you handle that depend on what you re uing if you have a convenient polynomial olver itting in front of you, there no problem with jut multiplying it all out and letting the olver olve it. If you re going to be doing it by hand, it may

8 require ome creative factoring, or, you might be better off doing ome Gauian elimination firt to introduce ome zero into the determinant before epanding. Keeping with the cofactor epanion though (and watch the grouping) det( AI) ( ) ( )[( )( ) ()] [( )( ) ()()] [( )() ( )()] ( )( 6) () ( 6 ) ( )( 6) 6( ) ( 5) At thi point, the characteritic polynomial i implified, and it fine to ue a olver (i.e. your calculator) to olve and get,,. I will alo note though that the above doe factor by grouping: ( ) ( ) ( )( ) ( )( )( ) Either way,,, are the eigenvalue of A. : rref From the firt row,. Let and o.

9 ,, i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion. : rref From the econd row,. Let and o.,, i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion. : ( ) ( ) ( ) 4 rref Let and o and.

10 ,, i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion. P5: Solve for eigenvalue: A AI det( AI) ( ) ( ) ( )[( )( ) ()] [( )( ) ()( )] [( )() ( )( )] ( )[( )( )] [ 6 4] [( )] ( )( )( ) ( ) 4( ) ( )( )( ) 4( ) 4( ) ( )( )( ) Notice with thi one I did ome creative grouping. It would alo be fine to multiply the whole thing out and ue a olver if you do, you ll be olving 7 5 It jut worth noting that if you do multiply it all the way out, there no way you ll be able to factor it. Either way, you ll get,, a the eigenvalue of A.

11 : rref From the firt row,. Let and o.,, i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion. : rref Let and o and,, i an eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion.

12 P6: 4 A 7 Solve for eigenvalue: 4 AI 7 det( AI) (4 ) ( ) 7 7 (4 )[( )(7 ) 4] [(7 ) 4] [ ( )( )] (4 )[4 9 4] [ 4 ] (4 )( 98) (6 ) Uing a polynomial olver give 7 and. : 4 7 rref 4 Let t and o t. t t t t t

13 ,, and,, are eigenvector of A, aociated with. The eigenpace of A aociated with i the pace generated by,, and,,. A bai for thi pace i {,,,,, }, and ha dimenion. 7 : rref 5.5 Let t and o t and t. t t t t,, i an eigenvector of A, aociated with 7. You could alo ue,, if you d like it to look cleaner. The eigenpace of A aociated with 7 i the pace generated by,,. A bai for thi pace i {,, }, and ha dimenion.