Re: EIGEN VALUES
Let me explain with an example.
Let V be a column vector ( 3 X 1). Each element in the column vector can be thought of as the componet of a "VECTOR" in three mutually perpendicular axes.
And A be a (3X3) matrix.
Now if i compute A*V we will get another column vector (3 x 1) say U
i.e. A*V = U
These V and U may or may not have same orientation or direction. They may or maynot have same magnitude. So because of this multiplication A*V, i am getting another vector U, which is a stretched (or compressed) and direction changed version of V.
If both U and V have the same orientation, then this multiplication A*V is equivalent to multiplication by a scalar (λ) as far as V is concerned, as the direction is unchanged. Then this particular vector V is called EIGEN VECTOR of matrix A. So for the EIGEN VECTOR V, A*V =λV. This scalar λ corresponding to EIGEN VECTOR V is called the EIGEN VALUE.
So for the EIGEN VECTOR V, multiplication by the matrix A can be replaced by multiplication by the scalar λ.
We can note that for each EIGEN VALUE there will be a number of EIGEN VECTORS, each one related to the other by a scale factor. Because of that EIGEN VECTORS are computed for unit magnitude.
EIGEN VECTORS and EIGEN VALUES are derived from the Matrix involved.
EIGEN VALUES have some properties
1. Sum of the EIGEN VALUES is equal to the sum of the elements in the main diagonal of A.
2.Product of EIGEN VALUES is equal to the Determinant A.
Read book by Lari Moore on Adaptive signal processing.
Cheers....