find all eigen value of matrix
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EIGEN VALUES AND EIGEN VECTORS
Consider multiplying a square 3x3 matrix by a 3x1 (column) vector. The result is a 3x1 (column) vector. The 3x3 matrix can be thought of as an operator - it takes a vector, operates on it, and returns a new vector. There are many instances in mathematics and physics in which we are interested in which vectors are left "essentially unchanged" by the operation of the matrix. Specifically, we are interested in those vectors v for which Av=kv where A is a square matrix and k is a real number. A vector v for which this equation hold is called an eigenvector of the matrix A and the associated constant k is called the eigenvalue (or characteristic value) of the vector v. If a matrix has more than one eigenvector the associated eigenvalues can be different for the different eigenvectors.
Geometrically, the action of a matrix on one of its eigenvectors causes the vector to stretch (or shrink) and/or reverse direction.
In order to find the eigenvalues of a nxn matrix A (if any), we solve Av=kv for scalar(s) k. Rearranging, we have Av-kv=0. But kv=kIv where I is the nxn identity matriX
So, 0=Av-kv=Av-kIv=(A-kI)v. This equation is equivalent to a homogeneous system of n equations with n unknowns. This equation has a non-zero solution for v if and only if the determinant det(A-kI) is zero. Thus, by finding the zeros of the polynomial in k determined by the characteristic equation det(A-kI)=0, we will have found the eigenvalues of the matrix A