Find the eigenvalues?

Question

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.)

Answer 1

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+1

Let us equate the determinant to zero to find eigenvalues.

(a-k)²+1=0

It is clear that two eigenvalues are complex, no matter what a.

 

a-k=±√-1=±i;

k=a±i;

 

k₁=a-i; k₂=a+i 

 

I hope this is as clear as it gets. If you have questions, please, ask.