Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
We now proceed to prove it. Let ~ A and ~B be two square matrices, each of order
n, which commute with each other, that is,
~A ~B ~B ~A ˆ‰~A; ~BŠ ˆ0:
First, let be an eigenvalue of ~ A with multiplicity 1, corresponding to the eigenvector
X, so that
~AX ˆ X:
…3:74†
Multiplying both sides from the left by ~B
Because ~B ~A ˆ ~A ~B, we have
~B ~AX ˆ ~BX:
~A… ~BX† ˆ… ~BX†:
Now ~B is an n n matrix and X is an n 1 vector; hence ~BX is also an n 1
vector. The above equation shows that ~BX is also an eigenvector of ~ A with the
eigenvalue . Now X is a non-degenerate eigenvector of ~A, any other vector which
is an eigenvector of ~A with the same eigenvalue as that of X must be multiple of X.
Accordingly
~BX ˆ X;
where is a scalar. Thus we have proved that:
If two matrices commute, every non-degenerate eigenvector of
one is also an eigenvector of the other, and vice versa.
Next, let be an eigenvalue of A ~ with multiplicity k. So A ~ has k linearly independent
eigenvectors, say X 1 ; X 2 ; ...; X k , each corresponding to :
~AX i ˆ X i ; 1 i k:
Multiplying both sides from the left by ~B, we obtain
~A… ~BX i †ˆ… ~BX i †;
which shows again that ~BX is also an eigenvector of ~ A with the same eigenvalue .
Cayley±Hamilton theorem
The Cayley±Hamilton theorem is useful in evaluating the inverse of a square
matrix. We now introduce it here. As given by Eq. (3.57), the characteristic
equation associated with a square matrix ~A of order n may be written as a polynomial
f …† ˆXn
iˆ0
c i ni ˆ 0;
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