Question

Find all distinct (real or complex) eigenvalues of A. Then find the basic eigenvectors of A corresponding to each eigenvalue.
For each eigenvalue, specify the number of basic eigenvectors corresponding to that eigenvalue, then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue.

A = 11 −10

17 −15

Number of distinct eigenvalues: ?

Number of Vectors: ?

? : {???}

Answer 1

We have A =

11	-10
17	-15

The eigenvalues of A are solutions to its characteristic equation det(A- λI₂)= 0 or, λ²+4λ+5 = 0 or, (λ+2+i)(λ-2-i)= 0. Hence, the eigenvalues of A are λ₁ = -2-i and λ₂ = -2+i. Further, the eigenvector of A corresponding to the eigenvalue -2-i is solution to the equation (A+(2+i)I₂)X= 0. To solve this equation, we have to reduce A+(2+i)I₂ to its RREF which is

1	-(1/17)(13-i)
0	0

Now, if X = (x,y)^T, then the equation (A+(2+i)I₂)X= 0 is equivalent to x –(y/17)(13-i) or, x = (y/17)(13-i) so that X = ( y/17)(13-i) ,y)^T = (y/17) (13-i ,17)^T. Hence the eigenvector of A corresponding to the eigenvalue -2-i is (13-i ,17)^T.

Similarly, the eigenvector of A corresponding to the eigenvalue -2+i is solution to the equation (A+(2-i)I₂)X= 0. To solve this equation, we have to reduce A+(2-i)I₂ to its RREF which is

1	-(1/17)(13+i)
0	0

Now, if X = (x,y)^T, then the equation (A+(2-i)I₂)X= 0 is equivalent to x –(y/17)(13+i) or, x = (y/17)(13+i) so that X = ( y/17)(13+i) ,y)^T = (y/17) (13+i ,17)^T. Hence the eigenvector of A corresponding to the eigenvalue -2-i is (13+i ,17)^T.

Number of distinct eigenvalues : 2

Number of eigenvectors: 2

eigenvalue -2-i ; eigenvector (13-i ,17)^T.

eigenvalue -2+i ; eigenvector (13+i ,17)^T.