Complex Eigenvalues. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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2 omplex Numbes When solving fo the oots of a quadatic equation, eal solutions can not be found when the disciminant is negative. In these cases we use complex numbes to wite down the solutions. Fo example, if we ty to solve x 2-2x+2=0 we un into touble: x 2 ( 2) 2 2() ()(2) x 2 2 x i With the definition i 2 =-, we can wite the solutions as OMPLEX numbes, with a REL pat and an IMGINRY pat. In geneal, complex numbes look like a+bi, whee a and b ae eal. We call a the eal pat and b the imaginay pat of the complex numbe. Evey complex numbe has a conjugate a-bi. Multiplying conjugates yields a eal numbe: (a+bi)(a-bi)=a 2 +b 2. This is a vey useful tick! Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

3 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 We will find the eigenvalues fo this matix. Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

4 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 We will find the eigenvalues fo this matix. 5 0 ( )(5 ) ( 2) ( 2) 2 2() i ()(7) Next we find eigenvectos fo the eigenvalues. Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

5 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 i ( i) 5 8 ( 0 i) 0 i 8 i 0 0 i 2 i 0 0 Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

6 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 i ( i) 5 8 ( 0 i) 0 i 8 i 0 0 i 2 i 0 0 t this point we can wite down the equation and find the vecto. We could do ow eduction but it s not eally necessay (multiplying one ow by the conjugate of its complex enty should make it clea that the ows ae multiples of each othe. Let s use the second ow: x ( i) x2 0 x ( i) x2 i Hee is an eigenvecto fo λ=+i Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

7 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 We could epeat the same wok to find the othe eigenvecto, but we don t need to. Since the e-values ae conjugates, so ae thei eigenvectos. i i i i Hee ae the eigenvalues with thei eigenvectos. t this point we could use ou diagonalization technique to obtain a diagonal matix with the (complex) eigenvalues on the diagonal, just like we did when the eigenvalues wee eal. Instead, we will get a matix that shows the otation and stetch that we mentioned peviously. Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

8 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 We fom matix P whose columns ae the eal and imaginay pats of the eigenvecto. P i i 0 Real 0 Imaginay i The book uses the e-value with the negative imaginay pat, so we will do the same. The pocess will also wok if you choose the othe e-value. When we use this matix to do a similaity tansfomation on the oiginal matix (just like we did fo diagonalization) we get a new matix that will have the stetch and otation popeties we ae looking fo. Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

9 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 P 0 P 0 Use the shotcut fo 2x2 invese matices Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

10 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 P 0 P 0 Use the shotcut fo 2x2 invese matices P P Fo eigenvalue a-bi, the matix will always take the fom a b b a Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

11 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 Matix is simila to Matix. It has the same eigenvalues. Matix splits into 2 pats. One is just a stetch (multiplication by the length of the eigenvalue). The othe pat is a otation. a b b 0 0 cos( ) sin( ) sin( ) cos( ) Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

12 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 Matix is simila to Matix. It has the same eigenvalues. Matix splits into 2 pats. One is just a stetch (multiplication by the length of the eigenvalue). The othe pat is a otation. a b b 0 0 cos( ) sin( ) In ou example, the eigenvalue is +i, so = 7 sin( ) cos( ) Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

13 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 Matix is simila to Matix. It has the same eigenvalues. Matix splits into 2 pats. One is just a stetch (multiplication by the length of the eigenvalue). The othe pat is a otation. a b b 0 0 cos( ) sin( ) In ou example, the eigenvalue is +i, so = sin( ) cos( ) Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB

14 When a matix has complex eigenvalues they will always come in conjugate pais. The effect of multiplying by the matix will be a stetch and a otation. The example below should help claify the situation: 8 5 Matix is simila to Matix. It has the same eigenvalues. Matix splits into 2 pats. One is just a stetch (multiplication by the length of the eigenvalue). The othe pat is a otation. a b b 0 0 cos( ) sin( ) In ou example, the eigenvalue is +i, so = cos( ) 7 sin( ) cos( ) 76 So matix will stetch a vecto by 7, and otate it counte-clockwise by 76. This will coespond to an outwad spial when we look at the effect of matix. Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB