Question

4 Solving Matrix Problems by Iteration 34. For a starting vector of p = 110, 10), compute p10) = A'p for each matrix A in Exercise 33 (use your representation of A found in Exercise 33)

Answer 1

33. following matrices compute the representation UD, U for the Whoes eigenvalues and largest eigenvector you asked were to determine. (a) 4 2 2 Here A = 4 IN od 2 2 А can be diagonalized if there exists an invertible and diagonal matrix o such that 'A= PDP"! matrixe P find eigenvalues of the matrix A 1A-1I) - 0

14-0) o co 2 12-) : . 14-^)(2-) - 0X250 -> (1-2)(1-4)=0 2,4 2=214. by given The eigenvalues of the matrix A are for eigenvectors = 2 V = 1 AV, = for x = 4 1 z مدیہ ܐ The eigenretors compose the columns of Matrix P. 1 ::. P= 1 1 1 1 :. The diagonal mata's D is composed of the eigenvalues. -'. 4 Now pi= adja

pl. 1 popt A = PDP 07 UDAU -1 1 A = 1 2 ] :) 1 1 0 4 1 3, U 어 to Da ASUDA U-1 (6) [34 3 4 het 1 A- 2 3 4 first we find eigenvalues of the Matrix A IA-111-0 (1-1) 2 1 1 1 14->) 12-5-220 2.-5-532 and A2 St 533 2 cross poing the eigenvalues the eigenrectors are.

for 1, 5-133 2 1 + 333 for 12 6 0.4574 pa -1 2 -1.4574 + 333 ☆ 1 1 1 1 O 5+533 2 Fo.5222 0.2389 py 0.5222 0.7611 Hare A = PDP- or| A = UD, UT

2 1 (0) 2:] 2 3 first we find eigenvalues of the matrix a Here A. 2 1 2 3 1 A-111=0 (2-1) | * 6 (3-1) ... 12-5x+ 4 20 (1-1)(1-4) A=1 , 4 1 .. The eigenalues of the matrix A are given by XF 1,4 for x=1 eigenvector Av=xU 1 *** for A2 = 4 0.5 Uz

Columns of matoin P The eigenvetors compose the .. 1 05 1 1 The diagonal matrix D is Composed of eigenvalues .. 0= TE :] o 4 Now oto 2/3 F-2/3 ES り3 2/3 A = UDU (d) 4 - 1 [1 2 first we find the eigenvalues of the matrix a TA-AF1 20 Here A = 4 -1 1 2 How 1A-111: 114-11 ro 1 12->)

.. (12-6X+9)=0 (4-3) 2= シ 3,3 :. The eigenvalues of the matrix À are given by : 3. for x=3 the eigen rector. v-1] for diagonalization, the nuber of eigenvetors must be equal to the matrix dimentions. dim is 2 which is less There ах е. 1 eigenrector and therefore the matoix cannot be diagonalized, than. + Not oliagonalizableli (34) for a Starting rector of P = [10, 10] Compute pllol =Aopt for each matrix A. cao A=U DA O Here A = 4 o 2 2

1 Az [4] [5] : دا رو/)4 وا 5 4 | | ہ اما 0 2 b حل .o) A = | 08341 44552 ها 1024 ام A09 : (° - 10 49 50 O شش lo 47552 1024 .( هم | 104 560 4150 10485 66 (2) 2. 1 آ 23 A = 0, un 2 حط Here A- 2 3

-1 F-2/3 A: 12 1/3 1 1 4 213 2/3 [: *** [**] [1: - 213 V3 2/3 2/3 10 34 952 6. 34 9 5 25 6 ggo50 6 990 51 (10) Alo pt To 3495 26 3495 25 6 ggo 51 6 g9050 vo), = 6990510 130010 13 98lolo So it is not calculate by. (D) is not diagonizable the's process - I A = 4 17 1 2 196830 10 25 587 9 А (1 96830 - log 7781

P"° - Ald pt ܦ 31 5 25 6 3 8 6 9 ܢ ]] ܠ ܐ |. ܐ 338 13 9 € $ 3 6 ܘܙܟ 5 4 4, ܘ 9 5 r 5 4 4, ܘ 9 5

. 2... (b) : 4 A- U DA UN 5-J33 6 10.5222 0 2389) 글 - + T 2 10 S+JIT 0 5222 1 0 761 2 1b JI 77 + JII Fo.s222 0.2389) A 6 110 | 0 S+J33 1 D.S222 0.71 ) 1 2. 69 720 50 47 83807 100 A' 1 52 4 [ 322 10 4 5 807 5 10 P = APpt 697 20 50 0 4 7 8 3807 1 Io 10 4 58075 1 24 183 2 117, CSR, so p10 254,999 570