7—Operators and Matrices 176multiple integrals with hopeless limits of integration, toss it out and start over. You may even find
problem7.18useful. Ans:ma^2 / 6
7.36 Show that the set of all 2 × 2 matrices forms a vector space. Produce a basis for it, and so what
is its dimension?
7.37 In the vector space of the preceding problem, the following transformation defines an operator.
f(M) =SMS−^1. ForS, use the rotation matrix of Eq. (7.13) and compute the components of this
operatorf. The obvious choice of basis would be matrices with a single non-zero element 1. Instead,
try the basisI,σx,σy,σz. Ans: A rotation by 2 αabout they-axis,e.g.f(~e 1 ) =~e 1 cos 2α−~e 3 sin 2α.
7.38 What are the eigenvectors and eigenvalues of the operator in the preceding problem? Now you’ll
be happy I suggested the basis that I did.
7.39 (a) The commutator of two matrices is defined to be [A,B]=AB−BA. Show that this
commutator satisfies the Jacobi identity.
[A,[B,C]]+[B,[C,A]]+[C,[A,B]]= 0
(b) The anti-commutator of two matrices is{A,B}=AB+BA. Show that there is an identity like
the Jacobi identity, but with one of the two commutators (the inner one or the outer one) replaced by
an anti-commutator. I’ll leave it to you to figure out which.
7.40 Diagonalize each of the Pauli spin matrices of problem7.20. That is, find their eigenvalues and
specify the respective eigenvectors as the basis in which they are diagonal.
7.41 What are the eigenvalues and eigenvectors of the rotation matrix Eq. (7.13)? Translate the answer
back into a statement about rotating vectors, not just their components.
7.42 Same as the preceding problem, but replace the circular trigonometric functions in Eq. (7.13)
with hyperbolic ones. Also change the sole minus sign in the matrix to a plus sign. Draw pictures of
what this matrix does to the basis vectors. What is its determinant?
7.43 Compute the eigenvalues and eigenvectors of the matrix Eq. (7.18). Interpret each.
7.44 Look again at the vector space of problem6.36and use the basisf 1 ,f 2 ,f 3 that you constructed
there. (a) In this basis, what are the components of the two operators described in that problem?
(b) What is the product of these two matrices? Do it in the order so that it represents the composition
of the first rotation followed by the second rotation.
(c) Find the eigenvectors of this product and from the result show that the combination of the two
rotations is a third rotation about an axis that you can now specify. Can you anticipate before solving
it, what one of the eigenvalues will be?
(d) Does a sketch of this rotation axis agree with what you should get by doing the two original rotations
in order?
7.45 Verify that the Gauss elimination method of Eq. (7.44) agrees with (7.38).
7.46 What is the determinant of a nilpotent operator? See problem7.34.
7.47 (a) Recall (or look up) the method for evaluating a determinant using cofactors (or minors). For
2 ×2, 3×3, and in fact, forN×N arrays, how many multiplication operations are required for this.
Ignore the time to do any additions and assume that a computer can do a product in 10 −^10 seconds.
How much time does it take by this method to do the determinant for 10×10, 20×20, and 30× 30