Hard question
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Suppose you have an n-by-n numerical matrix M.
Suppose that the determinant of M is not equal to 0.
(There is still a lot to talk about when the determinant = 0, but let's not start on that here).
There will be n vectors v_{i}, i = {1, 2, ..., n}, such that
M*v_{i} = e_{i}*v_{i},
where e_{i} is a number, M*v_{i} is the product of a matrix and a vector, and e_{i}*v_{i} is the product of a number and a vector.
The vectors v_{i} are called eigenvectors.
The numbers e_{i} are called eigenvalues.
You can do many wonderful things with the eigenvectors and the eigenvalues of a matrix. For example, you can form an orthogonal matrix O, that has the eigenvectors of M as column vectors, and use O (and its transpose) to diagonalize M. The resulting diagonal matrix will have the eigenvalues of M as elements.
I will not start on these here, because any textbook on the subject will tell you a lot more, and a lot better.