Question

LA class!!

1. Find a diagonlizing matrix P for the matrix A and write A in the form A = PDP-1 where D is a diagonal matrix. AE 5 -6 3 3 -4 3 0 0 2 Also, use the diagonalization of A to compute A8, A-8, and eA.

Answer 1

#1 The given matrix is A= 5-6 37 3-4 3 02 The charecteristic equation for A is 5- 3 3 -4-2 3 2-4 (=») {-(5-7)(4+2) + 18'} - O (Expanding the determinant along and colour Yow -- (25+5)-4-3-7) +18} =0 (2-8) (22-2-2) = 0 (2-x) { 2²-2X+X-2}=0 = (2-1) { *(4-2) +1(2-2)}=0 => (2-2)2(a+1) = 0 Therefore eigenvalues of the matrix A are x1=-1 and 22=2 Let 74 be an X= x2 eigen vector of A corresponding to the eigen value x = 713 Then AX = , X 24 5 -6 3 -4 3 3 az -12 2 524-672 +3:13 = -24 lie. G2-642 +3130 → 27,-272 +23-0 34-4824383=0-22 7 34-3x2 + 3430 24-22 tayo

and 223 = -2 23=0 Therefore (1) and (2) becomes 224-2712=0 & > 74-71220 24-X2=0 24 = 32 So If we take x2=1. Then 24=1 is an eigen vector con corresponding eigen value a=-1 to the T:] het 24 22 be an eigen vector of A Corresponding to the value 72=2. eigen Then 5-6 3 - 4 3 2 3 2 72 73 574-672 +373 = 274 344-632+ 37320 or, 24 - 2x2 + x3=0 38,-4X2 +383 = 222 ² 34-682 +3R3=0 244-2762 +23=0 and 22₂ =223 Therefore the system is equivalent to 24-272+23=0 24 = 22.-23 >

Let x2=C and 23=d shere cidER. Then x=zerd 20-d +d are two to the value 22=2 Therefore linearly corresponding independent eigen vectors of eigen Taking linearly independent eigen vectors [1]. [3] 11:2 the three we in columns and construct matrix P as - 2 P = 1 O 0 Now det p = =1(1-2) = -1 1 1 1 1 2 - Adj p = 1 1 p! det p (Adj p) = 1 1 Νσιο PAP = 5 31 3 - 4 3 - e 0 2 11 - 2 - 7 (2001 TO 0 =D (say)

PlAP=D A = PDP) where P = and Da (diagonal matrix) A = PDP => AS = (PDP")8 = (Dp1) (PDP:')-- (pop") 8 times (PP) op! PDP'p) D (P"p> P Dipl 8 2 -1 1-1 - 1 1 2 1 1 1 O 1 29 2 -1 28 1 [+1 +29 2-29 -1+28 2-28 -1 +27-28 -1 +28 28 0 511-510 255 255 -254 257 A o 256

A= PDP- >> (A)T = (PDP4)' (P+)*b'p? PDYP A-8 PCD') py - 2 2 - - [! У, з о Y28 17 Y₂8 : 157 |[ %8 1.A -1 + 2 - -100 -1 + 1 -1 27 28 - + 28 ㅗ 28 Now I+ A + A 3 = eA : eA = I+ CPDP) + = 68D84) + ši (PbPr)tn--- I + (PDPT) + Ź (PD?p') + $(PBF") = P(I+D+ e + R.) py = P eD py +

.eA= peppt Now eP = I+D+ 2 + ร + [] + OO 2 0 +(-1)+(ypt ...} {1+2+0. {rt2tonite To oo Therefore ea=peppy 2-1 1 -- 22 2e² = [ 0 e2 eh - eltze? zelze² - ette² -elte? zel_e² cette² O e²