(a) This matrix is not positive definite, see (b) below
(b) Eigenvalues are and , so the matrix is not positive definite
(c) Power method will converge to the dominant eigenvalue, which is .
(d) Since dominant eigenvalue is , and we know that the other eigenvalue is such that , we can make a shift by . Indeed, performing power iteration to the will converge to the nondominant eigenvalue.
(a) This matrix is not positive definite
(b) Using Rayleigh quotient we find eigenvalues , and .
(c) Power method will converge to the dominant eigenvalue and the vector, which will be a linear combination of two different eigenvectors corresponding to the dominant eigenvalue.
Cross multiplying by corresponding eigenvectors, we obtain
Performing power iteration on we will obtain the vector
(a) The power will fail since it will not be able to “decide” whether to converge to eigenvector corresponding to eigenvalue or . The resulting vector will therefore converge to the linear combination of the two eigenvectors corresponding to eigenvalue and , and will alternate its direction at each of the steps.
(b) Consider the matrix , and use power method and inverse power method to compute eigenvalues and .
(c) Using will converge to the maximum positive eigenvalue of matrix . The convergence will be very slow, the error will be multiplied by
(a) Eigenvalues are and . Power method will presumptuously converge to linear combination of eigenvectors corresponding to these eigenvalues and will alternate its direction.
(b) The equation (4.13) will reduce to
(c) When we choose , we are shifting eigenvalues from and to and . The eigenvalues of will be different for nonzero . Choose to find eigenvalue and corresponding eigenvector for the eigenvalue of . Choose to find eigenvalue and corresponding eigenvector to .
It is easier to work with or matrix first to gain intuition. For example, type in matlab
In what follow, we assume that is even.
To find eigenvalues, write . To calculate this determinant, expand it in the first raw. The second determinant should be expanded in the last raw. The result is
To calculate eigenvectors, write in components.
The eigenvectors corresponding to are
The eigenvector corresponding to is
Similarly, the eigenvector corresponding to is
(b) Power method will fail since four eigenvalues are very close to each other, and , which is very close to 1
(c) If you do a shift, and , then the dominant eigenvalue is
(d) The next biggest eigenvalue in the shifted matrix will be , so the error will be reduced each time by
(e) To converge to one can choose . Then the eigenvalue will be the dominant one, so power method will converge to it.