Phys 2920, Spring 2009
Problem Set #4
1. Find the eigenvalues and eigenvectors of the second Pauli matrix σyfrom the last problem
set.
2. Find eigenvalues and eigenvectors for the matrices
Lx=¯h
√2
0 1 0
1 0 1
0 1 0
Ly=¯h
√2
0−i0
i0−i
0i0
3. Use Maple or something suitable to find the eigenvalues and eigenvectors of
A=
2.5 1.6−4.8
4.2−3.1 2.1
−1.5−2.6 5.5
Take any one of the eigenvectors output by the program and by hand check that it is a
unit vector.
4. Let the matrix Abe given by
A= 5−7
2 3 !
Consider a scheme where the new basis vectors are
e0
1=e1+ 4e2e0
2= 3e1+ 10e2
Find the representation of Ain the new basis.
5. Find the similarity transformation which diagonalizes the matrix
A=
1−3 3
3−5 3
6−6 4
This problem is a bit different in that it has a repeated eigenvalue; but you can still find
three independent eigenvectors with which to construct S.
6. Find the eigenvalues and eigenvectors of the matrix
B=
−3 1 −1
−7 5 −1
−6 6 −2
Can Bbe diagonalized? (Can we construct a transformation matrix Sto make Bdiagonal?)
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