Question

3. Find all eigenvalues and eigenvectors of the matrix -20 2 -1 0 2 0-2 1

Answer 1

Solution: Given Matrix -2 o 2 1 let A= 2 -2 Characters tics polynomil will be given as |A-x1) = 0 2 2 -2-1 se (-2-7) (-x) (-2-x) - 0 +2(0-(-x)2) > (-2-7)(x) (2x) + 4x = 0 -> (+2-x)² - 4 -> ((2+x)? - 2²) -> (2+x-2) (2+x+2)=0 - - (1) (+4) = 0 = x=0,04 are t = -4 Eigen values x2=0 =0

Eigen vector for Eigen value a, = -4 (-2-(-4) o 2 2 -(-4) =o 1 y 2 0 -2-E4) Z Eigen vector 2 0 2 1 4 2 2 Performing now operation Az = Rz-R, R22 R2 + R / 2 4 o 0 Now mising R2 Q2/4 2 2 n 1/2 o" o 0 o Z 2x + 2z=0 fat -t/2 Eigen vector = [17 y + 2 = 0 let [2=t/ 2x = -2t x= -t7 Eigen = N vector 1/2 ar -t/2 comes ponely to + = -4

Now for d₂ = 3=0 -2-0 0 2 - 0 2 - 2-0 - 2 0 2 - 0 2 -2 using now operation R3 Rz TRI 2 0 Noo ] RI R / -2 [. ::] [:) - x + z = 0 x=2 let 1 they Eigen vecter for d=0 ها در Also, for d3=0 x 0 = for 3=0 Eign vector, Y Z