i need to find the eigenvalues and vectors of this matrix
[0 1]
[-1 0]
lamda - A gives me this:
[Lamda -1]
[1 Lamda]
Solve the det of this to get lamda squared +1
Only solution is imaginary, i and -i
The matrix of ilamda - A:
[i -1]
[1 i]
Not sure what to do here. Never worked with imaginary numbers in a matrix. Can someone show me how to solve this. Thanks
You "rref" with complex number exactly like with real numbers! Just remember to multiply and divide correctly!
But I would bother with "rref" here. Just use the definition of "eigenvalue": If is an eigenvalue of A, then there exist a non-zero vector, v, such that and the eigenvectors are the vectors satisfying that.
Since i is an eigenvalue, you must have
.
That gives the two equations y= ix and -x= iy which, since 1/i= -i, are really the same equation. From y= ix, we can write .
The eigenvectors corresponding to eigenvalue i are multiples of .
Now you find the eigenvectors corresponding to eigenvalue -i.