Eigenvalues and Eigenvectors. Eigenvalues and Eigenvectors. Initialization

Transcription

1 Eigenvalues and Eigenvecors Iniializaion "D; General::spellD; Eigenvalues and Eigenvecors We will now review some ideas from linear algebra. Proofs of he heorems are eiher lef as exercises or can be found in any sandard ex on linear algebra. We know how o solve n linear equaions in n unknowns. I was assumed ha he deerminan of he marix was nonzero and hence ha he soluion was unique. In he case of a homogeneous sysem AX, if de HaL, he unique soluion is he rivial soluion X. If de HaL, here exis nonrivial soluions o AX. Suppose ha de HaL, and consider soluions o he homogeneous linear sysem a, x + a, x + a, x + + a,n- x n- + a,n x n a, x + a, x + a, x + + a,n- x n- + a,n x n a, x + a, x + a, x + + a,n- x n- + a,n x n ª ª ª ª ª ª a n, x + a n, x + a n, x + + a n,n- x n- + a n,n x n b n A homogeneous sysem of equaions always has he rivial soluion x, x,..., x n. Gaussian eliminaion can be used o obain he reduced row echelon form which will be used o form a se of relaionships beween he variables, and a non-rivial soluion. Example. Find he nonrivial soluions o he homogeneous sysem

2 Eigen_vv.nb Soluion. x + x - x x + x + x 5 x + 4 x + x Use Gaussian eliminaion o eliminae x and he resul is x + x - x - x + x - 6x + 6x Since he hird equaion is a muliple of he second equaion, his sysem reduces o wo equaions in hree unknowns: x + x - x - x + x We can selec one unknown and use i as a parameer. For insance, le x ; hen he second equaion implies ha x and he firs equaion is used o compue x. Therefore, he soluion can be expressed as he se of relaions: x - x x or X x x x - -

3 Eigen_vv.nb We can fin he soluion by enering he equaions ino Mahemaica. eqns 8 x + x x, x + x + x, 5 x + 4x + x <; Idenify he marix of coefficiens A and column vecor B for he marix problem AX B.

4 Eigen_vv.nb 4 vars 8x,x,x <; Needs@"LinearAlgebra`MarixManipulaion`"D; mas LinearEquaionsToMarices@eqns, varsd; A mas PT ; B mas PT ; Prin@"Solve he equaions"d; Prin@TableForm@eqnsDD; Prin@"A ", MarixForm@ADD; Prin@"B ", MarixForm@BDD; Prin@"Solve he equaion A X B"D; Prin@" A ", MarixForm@AD, MarixForm@varsD, " ", MarixForm@BDD; Prin@"»A» ", De@ADD; Prin@"Hence he sysem will have non rivial soluions."d; Developer`LinearExpressionToMarix::obs : Developer`LinearExpressionToMarix has been superseded by CoefficienArrays, and is now obsolee. I will no be included in Mahemaica version 8. Solve he equaions x + x x x + x + x 5x + 4x + x A B 5 4 Solve he equaion A X B A»A» 5 4 x x x Hence he sysem will have non rivial soluions. Form he augmened marix M [A, B] and perform Gauss-Jordan eliminaion wih row inerchanges.

5 Eigen_vv.nb 5 b Pariion@B, D; M AppendRows@A, bd; m M; Prin@"A ", MarixForm@ADD; Prin@"B ", MarixForm@BDD; Prin@"The augmened marix is"d; Prin@"M ", MarixForm@MDD; A B 5 4 The augmened marix is M 5 4 Find he reduced row echelon form of he augmened marix M [A, B]. Prin@"The augmened marix is"d; Prin@"M ", MarixForm@MDD; The augmened marix is M 5 4 M PT M PT M PT ; M PT M PT 5 M PT ; Prin@MarixForm@MDD; 6 6 M PT M PT; Prin@MarixForm@MDD; 6 6

6 Eigen_vv.nb 6 M PT M PT M PT ; M PT M PT + 6 M PT ; Prin@MarixForm@MDD; The equaion form for his marix is x + x x - x There is one free variable which we choose o be x. I is used in compuing x and x -. The soluion vecor X x x x is X ; We are done. Aside. We can verify ha his is he soluion by direc muliplicaion A X. This is jus for fun!

7 Eigen_vv.nb 7 Prin@"Solve he equaion A X "D; Prin@" A ", MarixForm@AD MarixForm@varsD, " ", MarixForm@BDD; Prin@" X ", MarixForm@varsD, " ", MarixForm@XDD; Prin@"Does ", MarixForm@AD, MarixForm@XD, " ", MarixForm@BD, "?"D; Prin@" ", MarixForm@A.XD, " ", MarixForm@BDD; Prin@Flaen@A.XD Flaen@BDD; Solve he equaion A X A 5 4 x x x X x x x X Does 5 4 X 88,, <, 8,, <, 85, 4, <<.X 88,, <, 8,, <, 85, 4, <<.X 8,, <? Aside. We can le Mahemaica find he reduced row echelon marix. This is jus for fun! M m; Prin@"M ", MarixForm@MDD; Prin@"The row reduced echelon form is"d; Prin@" ", MarixForm@RowReduce@MDDD; M 5 4 The row reduced echelon form is Background for Eigenvalues and Eigenvecors Definiion (Linearly Independen). The vecors U, U,..., U n

8 Eigen_vv.nb 8 are said o be linearly independen if he equaion c U + c U c n U n implies ha c, c,..., c n. If he vecors are no linearly independen hey are said o be linearly dependen. Two vecors in are linearly independen if and only if hey are no parallel. Three vecors in are linearly independen if and only if hey do no lie in he same plane. Definiion (Linearly Dependen). The vecors U, U,..., U n are said o be linearly dependen if here exiss a se of numbers 8c, c,..., c n < no all zero, such ha c U + c U c n U n. Theorem. The vecors U, U,..., U n are linearly dependen if and only if a leas one of hem is a linear combinaion of he ohers. A desirable feaure for a vecor space is he abiliy o express each vecor as s linear combinaion of vecors chosen from a small subse of vecors. This moivaes he nex definiion. Definiion (Basis). Suppose ha S 8U, U,..., U m < is a se of m vecors in n. The se S i s called a basis for n if for every vecor X œ n here exiss a unique se of scalars 8c, c,..., c m < so ha X can be expressed as he linear combinaion X c U + c U c n U n

9 Eigen_vv.nb 9 Theorem. In n, any se of n linearly independen vecors forms a basis of n. Each vecor X œ n is uniquely expressed as a linear combinaion of he basis vecors. Theorem. Le K, K,..., K m be vecors in n. (i) If m>n, hen he vecors are linearly independen. (ii) If mn, hen he vecors are linearly dependen if and only if de HKL, where K,..., K m D. Proof Eigenvalues and Eigenvecors Eigenvalues and Eigenvecors Applicaions of mahemaics someimes encouner he following quesions: Wha are he singulariies of A -li, where l is a parameer? Wha is he behavior of he sequence of vecors 9A j X j? Wha are he geomeric feaures of a linear ransformaion? Soluions for problems in many differen disciplines, such as economics, engineering, and physics, can involve ideas relaed o hese equaions. The heory of eigenvalues and eigenvecors is powerful enough o help solve hese oherwise inracable problems. Le A be a square marix of dimension n n and le X be a vecor of dimension n. The produc Y AX can be viewed as a linear ransformaion from n-dimensional space ino iself. We wan o find scalars l for which here exiss a nonzero vecor X such ha () AX lx; ha is, he linear ransformaion T(X) AX maps X ono he muliple lx. When his occurs, we call X an eigenvecor ha corresponds o he eigenvalue l, and ogeher hey form he eigen-

10 Eigen_vv.nb pair l, A for A. In general, he scalar l and vecor X can involve complex numbers. For simpliciy, mos of our illusraions will involve real calculaions. However, he echniques are easily exended o he complex case. The n n ideniy marix I can be used o wrie equaion () in he form () HA -lil X. The significance of equaion () is ha he produc of he marix A -li and he nonzero vecor X is he zero vecor! The heorem of homogeneous linear sysem says ha () has nonrivial soluions if and only if he marix A -li is singular, ha is, () de HA -lil. This deerminan can be wrien in he form (4) a, -l a, a, a,n- a,n a, a, -l a, a,n- a,n a, a, a, -l a,n- a,n ª ª ª ª ª a n-, a n-, a n-, a n-,n- -l a n-,n a n, a n, a n, a n,n- a n,n -l Definiion (Characerisic Polynomial). When he deerminan in (4) is expanded, i becomes a polynomial of degree n, which is called he characerisic polynomial

11 Eigen_vv.nb (5) p HlL de HA -lil p HlL H-L n Il n + c l n- + c l n c n- l + c n- l + c n M Exploraion for p(λ) n ; A Table@a i,j, 8i, n<, 8j, n<d; I n IdeniyMarix@nD; p@λ_d De@A λi n D; solse Solve@p@λD, λd; Prin@" A ", MarixForm@ADD; Prin@"p@λD De@", MarixForm@A λi n D, "D"D; Prin@"p@λD ", Collec@Expand@p@λD p@dd, λd, "+H", Expand@p@DD, "L"D; Prin@"Solve p@λd and ge"d; Prin@TableForm@solseDD; A K a, a, O a, a, p@λd De@ λ + a, a, a, λ + a, D p@λd λ +λh a, a, L+H a, a, + a, a, L Solve p@λd and ge λ Ja, + a, a, + 4a, a, a, a, + a, N λ Ja, + a, + a, + 4a, a, a, a, + a, N

12 Eigen_vv.nb n ; A Table@a i,j, 8i, n<, 8j, n<d; I n IdeniyMarix@nD; p@λ_d De@A λi n D; Prin@" A ", MarixForm@ADD; Prin@"p@λD De@", MarixForm@A λi n D, "D"D; Prin@"p@λD ", Collec@Expand@p@λD p@dd, λd, "+H", Expand@p@DD, "L"D; A a, a, a, a, a, a, a, a, a, p@λd De@ λ + a, a, a, a, λ + a, a, D a, a, λ + a, p@λd λ +λ Ha, + a, + a, L + λ Ha, a, a, a, + a, a, + a, a, a, a, a, a, L+H a, a, a, + a, a, a, + a, a, a, a, a, a, a, a, a, + a, a, a, L

13 Eigen_vv.nb n 4; A Table@a i,j, 8i, n<, 8j, n<d; I n IdeniyMarix@nD; p@λ_d De@A λi n D; Prin@" A ", MarixForm@ADD; Prin@"p@λD De@", MarixForm@A λi n D, "D"D; Prin@"p@λD ", Collec@Expand@p@λD p@dd, λd, "+H", Expand@p@DD, "L"D; A a, a, a, a,4 a, a, a, a,4 a, a, a, a,4 a 4, a 4, a 4, a 4,4 p@λd De@ λ + a, a, a, a,4 a, λ + a, a, a,4 a, a, λ + a, a,4 a 4, a 4, a 4, λ + a 4,4 D p@λd λ 4 +λ H a, a, a, a 4,4 L + λ H a, a, + a, a, a, a, a, a, + a, a, + a, a, a,4 a 4, a,4 a 4, a,4 a 4, + a, a 4,4 + a, a 4,4 + a, a 4,4 L + λ Ha, a, a, a, a, a, a, a, a, + a, a, a, + a, a, a, a, a, a, + a,4 a, a 4, a, a,4 a 4, + a,4 a, a 4, a, a,4 a 4, a,4 a, a 4, + a, a,4 a 4, + a,4 a, a 4, a, a,4 a 4, a,4 a, a 4, a,4 a, a 4, + a, a,4 a 4, + a, a,4 a 4, + a, a, a 4,4 a, a, a 4,4 + a, a, a 4,4 + a, a, a 4,4 a, a, a 4,4 a, a, a 4,4 L +Ha,4 a, a, a 4, a, a,4 a, a 4, a,4 a, a, a 4, + a, a,4 a, a 4, + a, a, a,4 a 4, a, a, a,4 a 4, a,4 a, a, a 4, + a, a,4 a, a 4, + a,4 a, a, a 4, a, a,4 a, a 4, a, a, a,4 a 4, + a, a, a,4 a 4, + a,4 a, a, a 4, a, a,4 a, a 4, a,4 a, a, a 4, + a, a,4 a, a 4, + a, a, a,4 a 4, a, a, a,4 a 4, a, a, a, a 4,4 + a, a, a, a 4,4 + a, a, a, a 4,4 a, a, a, a 4,4 a, a, a, a 4,4 + a, a, a, a 4,4 L There exis exacly n roos (no necessarily disinc) of a polynomial of degree n. Each roo l can be subsiued ino equaion () o obain an underdeermined sysem of equaions ha has a corresponding nonrivial soluion vecor X. If l is real, a real eigenvecor X can be consruced. For emphasis, we sae he following definiions. Definiion (Eigenvalue). If A is and n n real marix, hen is n eigenvalues l, l,..., l n are he real and complex roos of he characerisic polynomial

14 Eigen_vv.nb 4 p HlL de HA -lil. Definiion (Eigenvecor). If l is an eigenvalue of A and he nonzero vecor V has he propery ha AV lv hen V is called an eigenvecor of A corresponding o he eigenvalue l. Togeher, his eigenvalue l and eigenvecor V is called an eigenpair l, V. The characerisic polynomial p HlL H-L n Il n + c l n- + c l n c n- l + c n- l + c n M can be facored in he form p HlL H-L n Hl-l L m Hl-l L m... Il -l j M m j... Hl -l mk- L m k- Hl-l mk L m k where m j is called he mulipliciy of he eigenvalue l j. The sum of he mulipliciies of all eigenvalues is n; ha is, n m + m m j m k- + m k. The nex hree resuls concern he exisence of eigenvecors. Theorem (Corresponding Eigenvecors). Suppose ha A is and n n square marix. (a) For each disinc eigenvalue l here exiss a leas one eigenvecor V corresponding o l. (b) If l has mulipliciy r, hen here exis a mos r lin-

15 Eigen_vv.nb 5 early independen eigenvecors V, V,..., V r ha correspond o l. Theorem (Linearly Independen Eigenvecors). Suppose ha A is and n n square marix. If he eigenvalues l, l,..., l k are disinc and l, V ; l, V ;...; l k, V k are he k eigenpairs, hen 8V, V,..., V k < is a se of k linearly independen vecors. Theorem (Complee Se of Eigenvecors). Suppose ha A is and n n square marix. If he eigenvalues of A are all disinc, hen here exis n nearly independen eigenvecors V, V,..., V n. Finding eigenpairs by hand compuaions is usually done in he following manner. The eigenvalue l of mulipliciy r is subsiued ino he equaion HA -lil V. Then Gaussian eliminaion can be performed o obain he row reduced echelon form, which will involve n-k equaions in n unknowns, where k r. Hence here are k free variables o choose. The free variables can be seleced in a judicious manner o produce k linearly independen soluion vecors V, V,..., V k ha correspond o l.

16 Eigen_vv.nb 6 NewonRaphson@x_, max_d : ModuleB8<, k ; p N@xD; Prin@"p ", PaddedForm@p, 86, 6<D, ", f@p D ", NumberForm@f@pD, 6DD; p p; WhileB k < max, p p; p p f@pd f'@pd ; k k + ; Prin@"p" k, " ", PaddedForm@p, 86, 6<D, ", f@", "p" k, "D ", NumberForm@f@pD, 6DD; F; Prin@" p ", NumberForm@p, 6D D; Prin@" p ±", Abs@p pd D; Prin@"f@pD ", NumberForm@f@pD, 6DD; F Example. Find he eigenvalues and eigenvecors of he marix A K O. Soluion. A ; I IdeniyMarix@D; M A λi ; p@λ_d De@MD; solse Solve@p@λD, λd; Prin@" A ", MarixForm@ A DD; Prin@" M ", MarixForm@ M DD; Prin@"p@λD De@MD ", p@λdd; Prin@"Solve p@λd ge"d; Prin@TableForm@solseDD; A K O M K λ λ O p@λd De@MD λ+λ Solve p@λd ge λ λ +

17 Eigen_vv.nb 7 i ; λ i ; B 8, <; b Pariion@B, D; Needs@"LinearAlgebra`MarixManipulaion`"D; M AppendRows@M, bd; Prin@"The augmened marix M is"d; Prin@"M" i, " ", MarixForm@MDD; M ReplaceAll@M, λ λ i D; M AppendRows@M, bd; Prin@"Subsiue ", "λ" i, " ", λ i D; Prin@"The augmened marix M is"d; Prin@"M" i, " ", MarixForm@MDD; Prin@"The row reduced echelon form is"d; Prin@MarixForm@ RowReduce@ MD DD The augmened marix M is M K λ λ O Subsiue λ The augmened marix M is M The row reduced echelon form is This is equivalen o he linear sysem x + x Se x and solve for x, and ge he eigenvecor V ; Verify he eigenpair.

18 Eigen_vv.nb 8 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i D, " ", ExpandAll@MarixForm@λ i V i DDD; Prin@ExpandAll@A.V i D ExpandAll@λ i V i DD; A K O λ V Does A V λ V? A V K O + + λ V H L J N True

19 Eigen_vv.nb 9 i ; λ i + ; B 8, <; b Pariion@B, D; Needs@"LinearAlgebra`MarixManipulaion`"D; M AppendRows@M, bd; Prin@"The augmened marix M is"d; Prin@"M" i, " ", MarixForm@MDD; M ReplaceAll@M, λ λ i D; M AppendRows@M, bd; Prin@"Subsiue ", "λ" i, " ", λ i D; Prin@"The augmened marix M is"d; Prin@"M" i, " ", MarixForm@MDD; Prin@"The row reduced echelon form is"d; Prin@MarixForm@ RowReduce@ MD DD The augmened marix M is M K λ λ O Subsiue λ + The augmened marix M is M The row reduced echelon form is This is equivalen o he linear sysem x x Se x and solve for x, and ge he eigenvecor V ; Verify he eigenpair.

20 Eigen_vv.nb ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i D, " ", ExpandAll@MarixForm@λ i V i DDD; Prin@ExpandAll@A.V i D ExpandAll@λ i V i DD; A K O λ + V Does A V λ V? A V K O + + λ V H + L J + N True Remark. Newon's mehod can be used o find he roos of he characerisic polynomial. For he firs eigenvalue. f@x_d p@xd; Prin@"p@xD ", p@xdd; NewonRaphson@.5, 4D; p@xd x+ x p.5, f@p D.5 p , f@p D p , f@p D p , f@p D p , f@p 4 D p p ± f@pd Which is an approximaion o he eigenvalue

21 Eigen_vv.nb " ", λ, " ", D,7DD; λ For he second eigenvalue. ", 6D; x+ x p.5, f@p D.75 p.5, f@p D.65 p , f@p D p , f@p D p , f@p 4 D p , f@p 5 D p , f@p 6 D. p p ±.587 f@pd. Which is an approximaion o he eigenvalue Prin@"λ", " ", λ, " ", NumberForm@N@λ D,7DD; λ Free Variables When he linear sysem is underdeermined, we needed o inroduce free variables in he proper locaion. The following subrouine will rearrange he equaions and inroduce free variables in he locaion hey are needed. Then all ha is needed o do is find he row reduced echelon form a second ime. This is done a he end of he nex example.

22 Eigen_vv.nb : ModuleA8c, i, k, L, M M, m, n, Z<, n Dimensions@MD PT ; m Dimensions@MD PT ; L 8, s, r, q, p, u, v, w, z, y, x<; Z Table@, 8m<D; c ; For@i, i n, i++, If@M Pi,iT, For@k n, i < k, k, M PkT M Pk T D; D; D; For@i n, i, i, If@M Pi,iT, M PiT Z; M Pi,iT ; M Pi,mT L PcT ; c c + ; D; D; Reurn@MDE Example. Find he eigenvalues and eigenvecors of he marix - A -. - Soluion. Find he characerisic polynomial and he eigenvalues.

23 Eigen_vv.nb A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; A A λi λ A λi λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 6 λ+6 λ λ p@λd H +λlh +λlh +λl To find he eigenvalues of he marix A Solve 6 λ+6 λ λ Le us plo p@λd 6 λ+6 λ λ and see where he roos are locaed

24 Eigen_vv.nb 4 Needs@"Graphics`Colors`"D Plo@p@λD, 8λ,,.5`<, PloSyle MagenaD Prin@"p@λD ", p@λdd; p@λd 6 λ+6 λ λ Alhough his example has been "cooked up" so ha he values are simple, we should be aware ha a roo finding mehod could be employed o find he eigenvalues. For illusraion, we can use he Newon-Raphson mehod. f@x_d p@xd;

25 Eigen_vv.nb 5 NewonRaphson@., 6D; p., f@p D p , f@p D p , f@p D p , f@p D p , f@p 4 D p , f@p 5 D p 6., f@p 6 D. p. p ±4. 4 f@pd. NewonRaphson@., 4D; p., f@p D.9 p , f@p D p.974, f@p D p , f@p D p 4., f@p 4 D. p. p ± f@pd. NewonRaphson@., 5D; p., f@p D p , f@p D p , f@p D p , f@p D p 4.96, f@p 4 D p , f@p 5 D p p ±9.6 f@pd

26 Eigen_vv.nb 6 Since we have solved for roos in previous modules, we will concenrae our effor on solving for he eigenvecors. Firs, we shall auomae he procedure for finding he roos of he characerisic polynomial, which is one way o find he eigenvalues. A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; Prin@"Ge"D; For@i, i n, i++, Prin@" ", "λ" i, " ", λ i, " ", Chop@N@λ i DDD D; A A λi λ A λi λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 6 λ+6 λ λ p@λd H +λlh +λlh +λl To find he eigenvalues of he marix A

27 Eigen_vv.nb 7 Solve 6 λ+6 λ λ Ge λ. λ. λ. Invesigae he eigen-pair l,v i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X x x x The augmened marix is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

28 Eigen_vv.nb 8 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

29 Eigen_vv.nb 9 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True Invesigae he eigen-pair l,v

30 Eigen_vv.nb i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X x x x The augmened marix is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

31 Eigen_vv.nb he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

32 Eigen_vv.nb ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True Invesigae he eigen-pair l,v

33 Eigen_vv.nb i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X The augmened marix M x x is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

34 Eigen_vv.nb 4 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

35 Eigen_vv.nb 5 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True The hree eigen-pairs are: Prin@"A ", MarixForm@ADD; For@i, i, i++, Prin@"λ" i, " ", λ i, ", ", "V" i, " ", MarixForm@V i DD; D; A λ, V λ, V λ, V We can compare his wih he resuls obained using Mahemai-

36 Eigen_vv.nb 6 ca's Eigensysem procedure. sol Eigensysem@AD; n Lengh@AD; Prin@"A ", MarixForm@ADD; For@i, i n, i++, Prin@"λ" i, " ", sol P,iT, ", ", "V" i, " ", MarixForm@sol P,iT DD; D; A λ, V λ, V λ, V Example 4. Find he eigenvalues and eigenvecors of he marix - A Soluion 4. Find he characerisic polynomial and he eigenvalues.

37 Eigen_vv.nb 7 A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; Prin@"Ge"D; For@i, i n, i++, Prin@" ", "λ" i, " ", λ i, " ", Chop@N@λ i DDD D; A A λi λ A λi λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 4 9 λ+6 λ λ p@λd H 4 +λlh +λl To find he eigenvalues of he marix A Solve 4 9 λ+6 λ λ Ge λ. λ. λ 4 4.

38 Eigen_vv.nb 8 Invesigae he eigen-pairs l,v and l,v. i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X The augmened marix M x x is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

39 Eigen_vv.nb 9 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables s Find he reduced row echelon form s s The eigenvecor is in he las column V s s The eigenvalue is repeaed, and here are wo linearly independen eigenvecors. Invesigae he eigen-pairs l,v and l,v. W V i ; Prin@"W ", MarixForm@WDD; W s s For V, se s in W and ge λ λ ; V i ReplaceAll@W, 8s <D; Prin@"V" i, " ", MarixForm@V i DD; V Verify he eigenpair.

40 Eigen_vv.nb 4 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True For V, se in W and ge i ; λ i λ ; V i ReplaceAll@W, 8 <D; Prin@"V" i, " ", MarixForm@V i DD; V s s Verify he eigenpair.

41 Eigen_vv.nb 4 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V s s Does A V λ V? A V s s s s λ V s s s s True Invesigae he eigen-pair l,v

42 Eigen_vv.nb 4 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ 4 Solve he equaion A X A X The augmened marix M x x is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

43 Eigen_vv.nb 4 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

44 Eigen_vv.nb 44 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ 4 V Does A V λ V? A V λ V 4 True I was good forune ha he hree eigenvecors are linearly independen. Prin@"A ", MarixForm@ADD; For@i, i, i++, Prin@"λ" i, " ", λ i, ", ", "V" i, " ", MarixForm@V i DD; D; A λ, V λ, V λ 4, V s s

45 Eigen_vv.nb 45 We can compare his wih he resuls obained using Mahemaicas Eigensysem procedure. sol n Prin@"A ", For@i, i n, i++, Prin@"λ" i, " ", sol P,iT, ", ", "V" i, " ", MarixForm@sol P,iT DD; D; A λ 4, V λ, V λ, V Example 5. Find he eigenvalues and eigenvecors of he marix A Soluion 5. Find he characerisic polynomial and he eigenvalues.

46 Eigen_vv.nb 46 A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; Prin@"Ge"D; For@i, i n, i++, Prin@" ", "λ" i, " ", λ i, " ", Chop@N@λ i DDD D; A A λi λ A λi λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 8 λ λ λ p@λd H +λli + λ+λ M To find he eigenvalues of he marix A Solve 8 λ λ λ Ge λ λ

47 Eigen_vv.nb 47 λ. Invesigae he eigen-pair l,v i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X + The augmened marix M x x is M + The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

48 Eigen_vv.nb 48 he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form J N J 5 N The eigenvecor is in he las column V J N J 5 N In his case he eigenvecor will have a nicer appearance if we replace wih. V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V H L H 5 L Verify he eigenpair.

49 Eigen_vv.nb 49 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V H L H 5 L Does A V λ V? A V λ V True H L H L H 5 L H L H 5 L H + 9 L H L H 9 L H + 9 L H L H 9 L Invesigae he eigen-pair l,v

50 Eigen_vv.nb 5 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ + Solve he equaion A X A X The augmened marix M x x is M + The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

51 Eigen_vv.nb 5 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form J 6 9 N J 5 + N The eigenvecor is in he las column V J 6 9 N J 5 + N In his case he eigenvecor will have a nicer appearance if we replace wih. V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V H 6 9 L H + 5 L Verify he eigenpair.

52 Eigen_vv.nb 5 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ + V H 6 9 L H + 5 L Does A V λ V? A V λ V True H + L H 6 9 L H + 5 L H 6 9 L H + 5 L H 9 L H 4 4 L H + 9 L H 9 L H 4 4 L H + 9 L Invesigae he eigen-pair l,v

53 Eigen_vv.nb 5 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x <; B 8,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X The augmened marix M x x is M + The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

54 Eigen_vv.nb 54 he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

55 Eigen_vv.nb 55 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True The hree eigen-pairs are: Prin@"A ", MarixForm@ADD; For@i, i, i++, Prin@"λ" i, " ", λ i, ", ", "V" i, " ", MarixForm@V i DD; D; A λ, V λ +, V H L H 5 L H 6 9 L H + 5 L λ, V We can compare his wih he resuls obained using Mahemaicas

56 Eigen_vv.nb 56 Eigensysem procedure. sol n Prin@"A ", For@i, i n, i++, Prin@"λ" i, " ", sol P,iT, ", ", "V" i, " ", MarixForm@sol P,iT DD; D; A λ +, V λ, V λ, V Example 6. Find he eigenvalues and eigenvecors of he marix - A. - Do his by consrucing he characerisic polynomial and finding is roos, and compare wih Mahemaica's Eigenvalues procedure. Soluion 6. Find he characerisic polynomial and he eigenvalues.

57 Eigen_vv.nb 57 A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; Prin@"Ge"D; For@i, i n, i++, Prin@" ", "λ" i, " ", λ i, " ", Chop@N@λ i DDD D; A A λi 4 λ A λi 4 λ λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 5 4 λ+7 λ 6 λ +λ 4 p@λd I λ+λ MI5 λ+λ M To find he eigenvalues of he marix A Solve 5 4 λ+7 λ 6 λ +λ 4 Ge

58 Eigen_vv.nb 58 λ J N λ J + N λ J N λ 4 J + N Invesigae he eigen-pair l,v i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4 <; B 8,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ J N Solve he equaion A X A X + J + N + J + N + J + N + J + N x x x x 4 The augmened marix is

59 Eigen_vv.nb 59 M + J + N + J + N + J + N + J + N The row reduced echelon form for M is + J + N Inroduce he free variables and find he eigenvecor. Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables + J + N Find he reduced row echelon form J + N J + N The eigenvecor is in he las column V J + J + N N In his case he eigenvecor will have a nicer appearance if we

60 Eigen_vv.nb 6 replace wih. V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V J + 4 J + N N Verify he eigenpair.

61 Eigen_vv.nb 6 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ J N J + N V 4 J + N Does A V λ V? A V J + 4 J + N N 4 J + 8 J + J + 4 J + N N N N λ V H J NL J + 4 J + N N J NJ + N J N J NJ + N J N True Invesigae he eigen-pair l,v

62 Eigen_vv.nb 6 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4 <; B 8,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ J + N Solve he equaion A X A X + J N + J N + J N + J N x x x x 4 The augmened marix is M + J + N + J + N + J + N + J + N The row reduced echelon form for M is

63 Eigen_vv.nb 6 J N Inroduce he free variables and find he eigenvecor. Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables J N Find he reduced row echelon form J + N J + N The eigenvecor is in he las column V J + N J + N In his case he eigenvecor will have a nicer appearance if we replace wih.

64 Eigen_vv.nb 64 V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V J + N 4 J + N Verify he eigenpair.

65 Eigen_vv.nb 65 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " H", λ i, "L", MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ J + N J + N V 4 J + N Does A V λ V? A V J + N 4 J + N 4 + J + N 8+ J + N + J + N 4+ J + N λ V H J + NL J + N 4 J + N J + NJ + N J + N J + NJ + N J + N True Invesigae he eigen-pair l,v

66 Eigen_vv.nb 66 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4 <; B 8,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ J N Solve he equaion A X A X + J + N + J + N + J + N + J + N x x x x 4 The augmened marix is M + J + N + J + N + J + N + J + N The row reduced echelon form for M is

67 Eigen_vv.nb 67 J + N Inroduce he free variables and find he eigenvecor. Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables J + N Find he reduced row echelon form J + N The eigenvecor is in he las column V J + N In his case he eigenvecor will have a nicer appearance if we replace wih. V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V J + N

68 Eigen_vv.nb 68 Verify he eigenpair. ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ J N V J + N Does A V λ V? A V J + N 6 J + 4 J + N N λ V J N J + N J NJ + N J N True Invesigae he eigen-pair l 4,V 4

69 Eigen_vv.nb 69 i 4; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4 <; B 8,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ 4 J + N Solve he equaion A 4 X A 4 X + J N + J N + J N + J N x x x x 4 The augmened marix M is M 4 + J + N + J + N + J + N + J + N The row reduced echelon form for M 4 is

70 Eigen_vv.nb 7 J N Inroduce he free variables and find he eigenvecor. Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables J N Find he reduced row echelon form J + N The eigenvecor is in he las column V 4 J + N In his case he eigenvecor will have a nicer appearance if we replace wih. V i V i ; Prin@"V" i, " ", MarixForm@V i DD; V 4 J + N Verify he eigenpair.

71 Eigen_vv.nb 7 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ 4 J + N V 4 J + N Does A V 4 λ 4 V 4? A V 4 J + N 6 + J + 4+ J + N N λ 4 V 4 J + N J + N J + NJ + N J + N True The four eigen-pairs are:

72 Eigen_vv.nb 7 Prin@"A ", MarixForm@ADD; For@i, i 4, i++, Prin@"λ" i, " ", λ i, ", ", "V" i, " ", MarixForm@V i DD; D; A λ J N, V J + 4 J + N N λ J + N, V λ J N, V λ 4 J + N, V 4 J + N 4 J + N J + N J + N We can compare his wih he resuls obained using Mahemaicas Eigensysem procedure.

73 Eigen_vv.nb 7 sol Eigensysem@AD; n Lengh@AD; Prin@"A ", MarixForm@ADD; For@i, i n, i++, Prin@"λ" i, " ", sol P,iT, ", ", "V" i, " ", MarixForm@sol P,iT DD; D; A λ J + N, V + J + N λ J N, V + J N λ J + N, V λ 4 J N, V J + + J N N Example 7. Find he eigenvalues and eigenvecors of he marix A. Soluion 7. Find he characerisic polynomial and he eigenvalues.

74 Eigen_vv.nb 74 A ; n Lengh@AD; I n IdeniyMarix@nD; M A λi n ; p@λ_d De@A λi n D; solse Solve@p@λD, λd; For@i, i n, i++, λ i solse Pi,,T ;D ; Prin@" A ", MarixForm@ADD; Prin@"A λ", "I" n, " ", MarixForm@AD, " λ", MarixForm@I n DD; Prin@"A λ", "I" n, " ", MarixForm@MDD; Prin@"The characerisic polynomial is"d; Prin@"p@λD»A λ I n»"d; Prin@"p@λD ", p@λdd; q@λ_d Facor@p@λDD; If@No@p@λD q@λdd, Prin@"p@λD ", q@λddd; Prin@"To find he eigenvalues of he marix A"D; Prin@"Solve ", p@λd D; Prin@"Ge"D; For@i, i n, i++, Prin@" ", "λ" i, " ", λ i, " ", Chop@N@λ i DDD D; A A λi 5 λ A λi 5 λ λ λ λ λ The characerisic polynomial is p@λd»a λ I n» p@λd 6 5 λ+56 λ 6 λ + λ 4 λ 5 p@λd H +λlh +λlh +λli 4 λ+λ M

75 Eigen_vv.nb 75 To find he eigenvalues of he marix A Solve 6 5 λ+56 λ 6 λ + λ 4 λ 5 Ge λ. λ. λ. λ λ Invesigae he eigen-pair l,v

76 Eigen_vv.nb 76 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4,x 5 <; B 8,,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X x x x x 4 x 5 The augmened marix is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

77 Eigen_vv.nb 77 he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

78 Eigen_vv.nb 78 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True Invesigae he eigen-pair l,v

79 Eigen_vv.nb 79 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4,x 5 <; B 8,,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X x x x x 4 x 5 The augmened marix is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

80 Eigen_vv.nb 8 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

81 Eigen_vv.nb 8 Prin@"A ", MarixForm@ADD; Prin@"λ" i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True Invesigae he eigen-pair l,v

82 Eigen_vv.nb 8 i ; A i ReplaceAll@M, λ λ i D; vars 8x,x,x,x 4,x 5 <; B 8,,,, <; Needs@"LinearAlgebra`MarixManipulaion`"D; M i AppendRows@A i, Pariion@B, DD; RM i RowReduce@ M i D; Needs@"LinearAlgebra`MarixManipulaion`"D; Prin@"For he eigenvalue ", "λ" i, " ", λ i D; Prin@"Solve he equaion ", "A" i, " X "D; Prin@"A" i, " X ", MarixForm@A i D MarixForm@varsD, " ", MarixForm@BDD; Prin@"The augmened marix ", "M" i, is"d; Prin@" ", "M" i, " ", MarixForm@M DD; Prin@"The row reduced echelon form for ", "M" i, " is"d; Prin@" ", MarixForm@ RM i DD; For he eigenvalue λ Solve he equaion A X A X The augmened marix M x x x x 4 x is M The row reduced echelon form for M is Inroduce he free variables and find he eigenvecor.

83 Eigen_vv.nb 8 Prin@"Inroduce he free variables"d; FM i FreeVariables@RM i D; Prin@MarixForm@FM i DD; Prin@"Find he reduced row echelon form"d; SM i RowReduce@ FM i D; Prin@MarixForm@SM i DD; Prin@"The eigenvecor is in he las column"d; V i TakeColumns@SM i, D; Prin@"V" i, " ", MarixForm@V i DD; Inroduce he free variables Find he reduced row echelon form The eigenvecor is in he las column V Verify he eigenpair.

84 Eigen_vv.nb 84 ", i, " ", λ i D; Prin@"V" i, " ", MarixForm@V i DD; Prin@"Does ", "A ", "V" i, " ", "λ" i, "V" i, "?"D; Prin@"A ", "V" i, " ", MarixForm@AD, MarixForm@V i D, " ", MarixForm@A.V i DD; Prin@"λ" i, "V" i, " ", λ i, MarixForm@V i D, " ", MarixForm@λ i V i DD; Prin@ExpandAll@A.V i λ i V i DD; A λ V Does A V λ V? A V λ V True Invesigae he eigen-pair l 4,V 4