Find the eigenvalues in terms of α for x'=(α, -1, 1, α)x. (this is 2x2 matrix, α and -1 on the left, 1 and α on the right.)

In order to find eigenvalues, you need to solve the secular equation:

det|A-kI|=0; here A is your matrix, A={(a,-1);(1,a)}, I is the unit matrix, {(1,0);(0,1)}, k is the sought eigenvalue.

Matrix A-kI looks as follows:

A-kI={(a-k,-1);(1,a-k)} (a-k and -1 on the left, 1 and a-k on the right)

Its determinant is:

(a-k)

^{2}+1Let us equate the determinant to zero to find eigenvalues.

(a-k)

^{2}+1=0It is clear that two eigenvalues are complex, no matter what a.

a-k=±√-1=±i;

k=a±i;

**k**

_{1}=a-i; k_{2}=a+iI hope this is as clear as it gets. If you have questions, please, ask.