CS 417/517 Computational Methods and Software
Spring 2004
Assignment 8
Assigned: Thurs April 8, 2004; Due: Thurs April 15, 2004
1. True or false? Give a reason for your answer to receive any credit.
(a) False, because eigenvalues are the roots of the charactarstic polynomial.
(b) False, n×nmatrix has nlinearly independent eigenvectors if it is not defective.
(c) True, If a square matrix is singular, then one of its eigenvalues is
equal to 0.
(d) False, If two matrices are similar, then they have the same eigenvalues.
(e) False, If two matrices A, B have same eigenvalues, and there exist a nonsingular matrix
Tsuch that B=T−1AT then they are similar.
2. The same eigenvalue correspond to two distinct eigenvectors, but
the same eigenvector can not correspond to two distinct eigenvalues.
3. (a) Hv = (I−2vTv)v
Hv =v−2vTvv =−v.
(b) Hx = (I−2vTv)x
Hx =x−2vTxv =x.
4. Yes, because λ3= 0
5. (a) λ2−2λ−3
(b) λ2−2λ−3 = 0
λ1= 3 and λ2=−1
(c) λ1= 3 and λ2=−1
(d) eigenvectors of the matrix
v1= 2
1!, v2= −2
1!.
(e)
1 4
1 1 ! 1
1!= 5
2!.
5/√29
2/√29 !
(f) Power iteration converge to the dominant eigenvalue.