Attachment 23173 Here is a problem that I have no idea how to solve. So can I ask you for help?
And since I am a beginner, I would be grateful if you could give me a source, from which I can read more about solving such type of problems.
Thank you very much in advance!
Jan 3rd 2012, 07:39 AM
You have no idea how to solve this? Even if you are a beginner, if you are given a problem like this, you should know how to find eigenvalues- at least the definition of "eigenvalue".
A number, , is said to be an "eigenvalue" of linear transformation A if and only if there exist a non-zero ("non-trivial") vector, v, satisfying . Obviously, v= 0 always satisfies that- the key here is "non-zero".
That equation is the same as . Now, in terms of matrices, if had an inverse, we could multiply on both sides by that inverse, showing that v= 0, the "trivial" solution is the only solution.
So is an eigenvalue of A if and only if does not have an inverse- and that is true if and only if the determinant of (written as a matrix) is 0. That is, any eigenvalue, must satisfy the "characteristic equation"
The "characteristic equation is .
That is a quadratic equation to be solved for . Using the quadratic formula to solve it will let you determine what values of give two real, one real, or two complex solutions.