Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
Example 3.15
Show that similar matrices have the same characteristic polynomial and hence the
same eigenvalues. (Another way of stating this is to say that the eigenvalues of a
matrix are invariant under similarity transformations.)
Solution: Let ~A and ~B be similar matrices. Thus there exists a third matrix ~S
such that ~B ˆ ~S 1 ~A ~S. Substituting this into the characteristic polynomial of
matrix ~B which is j ~B ~Ij, we obtain
j ~B Ij ˆj~S 1 ~A ~S ~Ij ˆj~S 1 … ~A ~I† ~Sj:
Using the properties of determinants, we have
j ~S 1 … ~A ~I† ~Sj ˆj~S 1 jj ~A ~Ijj ~Sj:
Then it follows that
j ~B ~Ij ˆj~S 1 … ~A ~I† ~Sj ˆj~S 1 jj ~A ~Ijj ~Sj ˆj~A ~Ij;
which shows that the characteristic polynomials of ~A and ~B are the same; their
eigenvalues will also be identical.
Eigenvalues and eigenvectors of hermitian matrices
In quantum mechanics complex variables are unavoidable because of the form of
the SchroÈ dinger equation. And all quantum observables are represented by hermitian
operators. So physicists are almost always dealing with adjoint matrices,
hermitian matrices, and unitary matrices. Why are physicists interested in hermitian
matrices? Because they have the following properties: (1) the eigenvalues of a
hermitian matrix are real, and (2) its eigenvectors corresponding to distinct eigenvalues
are orthogonal, so they can be used as basis vectors. We now proceed to
prove these important properties.
(1) the eigenvalues of a hermitian matrix are real.
Let ~H be a hermitian matrix and X a non-trivial eigenvector corresponding to the
eigenvalue , so that
~HX ˆ X:
…3:62†
Taking the hermitian conjugate and note that ~H y ˆ ~H, we have
X y ~H ˆ *X y :
…3:63†
Multiplying (3.62) from the left by X y , and (3.63) from the right by X y , and then
subtracting, we get
… *†X y X ˆ 0:
Now, since X y X cannot be zero, it follows that ˆ *, or that is real.
128
…3:64†