explain the eigenvalue problem

The eigenvalue problem is how we can find non-trivial solutions where x does not equal zero to the matrix equation;AX=LX (L=lambda)Values of the scalar L for which non-trivial solutions exist are called eigenvalues and the corresponding solutions of X where X does not equal 0 are called eigenvectors. A is an nn matrix.X is an n1 column vector.
We can write the above matrix equation, which represents the set of simultaneous equations as;(LI-A)X=0Where I is the identity matrix.This matrix equation represents a set of homogenous equations, thus we know that a non-trivial solution exists if the determinant of (LI-A) is equal to zero. The polynomial equal to the expansion of this determinant is called the characteristic equation of A, from which we find the eigenvalues and thus the eigenvectors.