Find an explicit 3 times 3 matrix A that has eigenvectors V1

Find an explicit 3 times 3 matrix A that has eigenvectors V_1 = (2, -1, 1) and V_2 = (1, 0, 1) with associated eigenvalues lambda_1 = 3 and lambda_2 = -2, respectively; N(A) = span((l, 1, 1)) and verify that your choice of A works.

Solution

We know that for a matrix A having an eigenvector v corresponding to the eigenvalue , we have Av = v. Let A=

a₁

b₁

c₁

a₂

b₂

c₂

a₃

b₃

c₃

Here, Av₁= 3v₁=3(2,-1,1)=(6,-3,3).Also,Av₁=(2a₁-b₁+c₁,2a₂-b₂+c_2,2a₃-b₃+c₃).Hence, 2a₁-b₁+c₁ = 6…(1),              2a₂-b₂+c₂ = -3…(2)_, and 2a₃-b₃+c₃ = 3…(3).

Similarly, Av₂= 2v₂=2(1,0,1)=(2,0,2). Also,Av₂=(a₁-c₁,a₂-c_2,a₃-c₃).Hence, a₁-c₁ = 2…(4),a₂-c₂ = 0…(5)_, and           a₃-c₃= 2…(6).Then, from the 4^th, 5^th and the 6^th equations, we have c₁ = a₁-2, c₂ = a₂ and c₃ = a₃+2. Also, on substituting these values of c₁,c₂,c₃ in the 1^st , 2^nd and the 3^rd equations, we have 2a₁-b₁+a₁ -2= 6   or, b₁ = 3a₁-8, 2a₂-b₂+a₂ = -3 or, b₂= 3a₂+3, 2a₃-b₃+a₃+2 = 3 or, b₃ = 3a₃-1.

Further, N(A) = span{(1,1,1)}. if X N(A), then AX = 0. Thus, if X = (1,1,1), then a₁+b₁+c₁ = 0, or, a₁+3a₁-8+a₁-2=0or, 5a₁=10 so that a₁ = 2. Similarly, a₂+b₂+c₂= 0 or, a₂+3a₂+3+a₂=0 or, 5a₂ = -3 so that a₂ = -3/5 and a₃+b₃+c₃ = 0 or, a₃+3a₃-1+a₃+2= 0 or, 5a₃= -1 so that a₃ = -1/5. Hence, A =

2

-2

0

-3/5

6/5

-3/5

-1/5

-8/5

9/5

On verification, we have Av₁= (6,-3,3) Av₂= (2,0,2) and A(1,1,1) = (0,0,0).Hence A satisfies all the required criteria.

a1	b1	c1
a2	b2	c2
a3	b3	c3