Description
1. Let
A =
1 1 1
0 1 1
s 0 2
, with s = −10−6
.
(a) Compute all the eigenvalues λi and corresponding right eigenvectors xi and left eigenvectors yi
.
(b) Compute the “overall condition number” for this eigenproblem based on the perturbation formula
|λA+E − λA| ≤ kXk · kX−1
k · kEk,
where X is the matrix of right eigenvectors and E is some generic perturbation matrix.
(c) Compute the condition numbers for every individual eigenvalue based on the perturbation approximation (ignoring higher order terms)
|λA+E − λA| ≈ O(kEk)
cos θ
=
kyik2kxik2
y
T
i xi
O(kEk),
where yi
, xi are the left and right eigenvectors corresponding to the eigenvalue λA for the matrix
A. Do this for each eigenvalue of A.
(d) Compute the eigenvalues of the matrix Ae defined as the same as the matrix A above but with
s = 0. How do the eigenvalues of Ae compare with what would be expected based on the condition
number bounds?
2. Compute all the eigenvalues and eigenvectors of the complex symmetric matrix
A =
2i 1
1 0!
3. Consider the block upper triangular matrix
A =
A11 A12
0 A22!
.
(a) Suppose A11u = λu, but λ is not an eigenvalue of A22. Find a vector v (in terms of Aij , u) such
that the vector
u
v
!
is an eigenvector of A. What is the corresponding eigenvalue?
(b) Suppose A22v = λv, but λ is not an eigenvalue of A11. Find a vector u (in terms of Aij , v) such
that the vector
u
v
!
is an eigenvector of A. What is the corresponding eigenvalue?
(c) Repeat the above assuming λ is an eigenvalue for both A11 and A22. Do any of the above cases
fail?
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