Question

Find the eigenvalues and eigenvectors of the matrix 6 5 B- [ -5-2

Answer 1

Solution : is The given matrix 5 B 6 5 -2 for to solve we the eigenvalues have characteristics equation, det (B-212) 6-7 5 => = O -5 -2-7 (9-) (2+2) +25 = 0 => 22-47 +13 = 0 4 + J16-52 ♡ T 2 2 + 3 i n a = 3-31, 22=2+31 i' The eigenvalues are ч be the eigenveetor corresponding ut X = 12 to the eigenvalue 21 =2-31 АХ nix А => 2. Iz) x FO

4t3 ( )) (9) - 나 + 3 21 -3i)지+572 G (+3i) 지 -SA + (4+i2 = 0 augmented matrix { i3 The 이 443 5 - 디 +3 4- 31 25 메 T 4-31 。。 5 - 다 +) R2 +5R 4-31 0 - 0 The system (i) is equivalent to 시+ 4일 12 - 2 0 - 4 + - 키 제 지 - J

-4t31 X2 X 5 22 ut X2 C (any Constant Perand not equal to o number then 4 +31 the is X = ☺ (4+0) Ci 5 1 2,-2-31 eigen vector Corresponding to 24 be the ut X= eigenvector Grresponding the eigenvalue 72 = 2t3i ( ) ใน to AX: 72 X -) (A - Az I») x = 0 4-3i 7 -4-3i +5X2 = 0 al (4-3i by - 504 + (-4-3i)*2 = 0 زل

The matrix is, augmented 4-3i 5 -5 -4-3, 4+1 RI 25 1 4+31 4+31 0 R2 +5R 1 -4-31 - Г O o The system o is equivalent to, 4+31 X2 = 0 rut - 21 -4-3i 22 5 4-3i x2 x = J x2 any constant 4-31 5 the X = e 2 et H2 = C2(+0) be (h (ato) is eigenvector Corresponding to the eigenvalue 2+31 72 -

The eigen eigenvalues and eigen vectors of the matrix are <-4+31 5 for X= eigen vector is 21=2-31 670] for 72 = 2+3; is X = (2 -4-31 J (2+0]