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# Class Note for MATH 250A at UA 2

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This 92 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Examples 0 O O Ojdetermilr12ur1tOgtltO OgtltOO r and Cl 02 must satisfy Eigenvalue Eigenvector Method xn1 all 12 xn yn1 6 21 6 22 yn n1 n n n1 n n for all n 02739 6121027 6122627 n1 n n C17 6111617 6112021 71 1 n n czr a2102r a2202r Eigenvalue Eigenvector Method xnl all an xquot y n1 6 21 6 22 yquot n1 n n r and Cl 02 clr 511101 alzczr must satisf n1 n n for all n y 027 612102 6122627 Cancel r CI 611161 611262 Czr 612162 612262 Eigenvalue Eigenvector Method xn1all a12 xn yn1 6 21 6 22 yn n1 n n r and Cl 02 C17 5111017 6112027 f M n n for all n must satls y czr 6121027 azzczr Cancel rquot all 7061 611262 2 0 and gather terms 612162 6122 I 62 0 r and Cl 02 must satisfy Eigenvalue Eigenvector Method xn1 all a12 xn y n1 6 21 6 22 yn n1 n n n1 n n for all n 027 6121027 azzczr a11 rc1 611262 2 0 612162 6122 rc2 0 Two equations two unknowns Cl 02 Eigenvalue Eigenvector Method xnlall a12 xn ynl 6 21 6 22 yn n1 n n r and Cl 02 01 5111017 6112027 for all n must satISf quot1 n n 39 y 027 6121027 azzezr 0 612162 6122 re2 0 a11 r 31 611262 Equations have only the solution 01 0 c2 O unless the determinant equals 0 all ra22 ra21a12 0 r and Cl 02 must satisfy Cancel r and gather terms Eigenvalue Eigenvector Method u a11 a12 xn ynl 6 21 6 22 yn qr 1 anclr alzczr n1 n n for all n 021 a czr azzczr a11 re1 611262 2 0 612162 6122 re2 0 Equations have only the solution 01 0 c2 O unless the determinant equals 0 all ra22 ra21a12 0 Pick r as a root of this equation then solve for Cl 02 Eigenvalue Eigenvector Method xn1all a12xn yn1 a2 6122 yn xquot c rquot j 1 j is a solution if r satis es the quadratic equation n yquot 021 all ra22 ra21a12 0 and Cl 02 satisfy the equations a11 rc1 611262 2 0 612162 6122 rc2 0 Eigenvalue Eigenvector Method xn1all a12 xn y n1 6 21 6 22 yn xi 01F n is a solution if r satis es the quadratic equation yn 627 2 r a11 6122 allaZZ a21a12 0 and c1 c2 satisfy the equations all rc1a2c2 0 a21c2a22 rc2 0 A quadratic equation the characteristic equation Eigenvalue Eigenvector Method xnlall a12 xn y n1 6 21 6 22 yn Xquot or n is a solution if r satis es the quadratic equation yn 627 2 r a11 6122 allaZZ a21a12 0 and Cl 02 satisfy the equations a11 rc1 611262 2 0 612162 6122 rc2 0 Recall we need solve only one of these two equations Eigenvalue Eigenvector Method xnlall a12 xn y n1 6 21 6 22 yn Xquot or n is a solution if r satis es the quadratic equation yn 627 2 r a11 6122 allaZZ a21a12 0 and Cl 02 satisfy the equations a11 rc1 611262 2 0 612162 6122 rc2 0 r is called an eigenvalue of the matrix c 0 1 7t 0 is called an eigenvector associated with r 02 Eigenvalue Eigenvector Method xnlall a12 xn y n1 6 21 6 22 yn Xquot or n is a solution if r satis es the quadratic equation 02 2 r a11 6122 allaZZ a21a12 0 and Cl 02 satisfy the equations a11 rc1 611262 2 0 612162 6122 rc2 0 r is called an eigenvalue of the matrix c 0 They bmld 1 7 j is called an eigenvector associated With r solutions 02 0 Population DVnamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 previous example 261 14 c yquot 08gtlt14quot c 2 any number Population Dvnamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example Cl11 I 61 611262 2 0 Xquot 14quot azlcz 6 22 7 Cz Z 0 C yquot 08gtlt14quot c 2 any number Population Dvnamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example 61 1562 0 xquot 0862 12CZ 0 C yquot 08 gtlt14 c 2 any number Population Dynamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example Cl 1502 2 0 QeSH lw250 xquot 14 2 2 C yquot 08 gtlt14 Need solve only one equation 6 2 any number Population Dynamic Application arm 2 02x 15yn xquot clr y 1 08x 04yn yquot 02739 r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example 61 1562 0 Jc 04nl W1 0 2 any number C 2 any real number Population Dynamic Application arm 2 02x 15yn xquot clr ynl 08er 0 4yn 341 Czrn r2 06r1120 roots rl4 Eigenvalue r 14 Eigenvalu r O8 previous example 61 l 2 0 Jc 04nl CH c 2 any number C 2 any real number Population Dynamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example x 15 gtlt 08n n C xquot 14 3 n yquot 08 gtlt14 c any number 6 2 any number Population Dynamic Application arm 2 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example x 15X 1n X 08quot xquot 14 yn 1 gtlt08 My CO8gtlt14n c 2 any number 6 2 any number Population Dynamic Application arm 2 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example x 15X 0 8quot xquot 14 3 08quot c yquot 08 gtlt14 C 2 any number 6 2 any number Population Dynamic Application an 02x 15yn xquot clr y 1 08x 04yn yquot czrn r206r1120 roots rl4 and 08 Eigenvalue r 14 Eigenvalue r O8 previous example x 15X 0 8n n 1quot6 3 08quot xquot 14 c yquot O8gtlt14 C 2 any num er 6 2 any fmber These solutions are unbounded These tend to 0

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