ABSTRACT
Both of the cases described above are solved by shifting the matrix by the largest (in absolute value) eigenvalue and applying the direct power method to the shifted matrix. Generally speaking, it is not known a priori which result will be obtained. If all the eigenvalues of a matrix have the same sign, the smallest (in absolute value) eigenvalue will be obtained. If a matrix has both positive and negative eigenvalues, the largest eigenvalue of opposite sign will be obtained,
The above procedure is called the shifted direct power method. The procedure is as follows:
I. Solve for the largest (in absolute value) eigenvalue ALargesl' 2. Shift the eigenvalues of matrix A by s = ALargeSl to obtain the shifted matrix
Ashifted' 3. Solve for the eigenvalue Ashifted of the shifted matrix Ashifted by the direct power
method. 4. Calculate the opposite extreme eigenvalue of matrix A by), = Ashifted + S.
