Mathematical Methods for Physicists: A concise introduction - Site Map

THE MATRIX EIGENVALUE PROBLEM
9
… a 11 †x 1 ‡ a 12 x 2 ‡ ‡a 1n x n ˆ 0
a 21 x 1 ‡ … a 22 †x 2 ‡ ‡a 2n x n ˆ 0
>=
...
>;
a n1 x 1 ‡ a n2 x 2 ‡ ‡… a nn †x n ˆ 0
…3:55†
In order to have a non-zero solution, we recall that the determinant of the coecients
must be zero; that is,
a 11 a 12 a 1n
a 21 a 22 a 2n
det… ~A ~I† ˆ
.
.
. . ˆ 0: …3:56†
a n1 a n2 a nn
The expansion of the determinant gives an nth order polynomial equation in ,
and we write this as
c 0 n ‡ c 1 n1 ‡ c 2 n2 ‡ ‡c n1 ‡ c n ˆ 0;
…3:57†
where the coecients c i are functions of the elements a jk of ~ A. Eq. (3.56) or (3.57)
is called the characteristic equation corresponding to the matrix ~A. We have thus
obtained a very important result: the eigenvalues of a square matrix ~A are the
roots of the corresponding characteristic equation (3.56) or (3.57).
Some of the coecients c i can be readily determined; by an inspection of Eq.
(3.56) we ®nd
c 0 ˆ…1† n ; c 1 ˆ…1† n1 …a 11 ‡ a 22 ‡‡a nn †; c n ˆ det ~ A: …3:58†
Now let us rewrite the characteristic polynomial in terms of its n roots
1 ; 2 ; ...; n
c 0 n ‡ c 1 n1 ‡ c 2 n2 ‡ ‡c n1 ‡ c n ˆ … 1 † … 2 †… n †;
then we see that
c 1 ˆ…1† n1 … 1 ‡ 2 ‡‡ n †; c n ˆ 1 2 n : …3:59†
Comparing this with Eq. (3.58), we obtain the following two important results on
the eigenvalues of a matrix:
(1) The sum of the eigenvalues equals the trace (spur) of the matrix:
1 ‡ 2 ‡‡ n ˆ a 11 ‡ a 22 ‡‡a nn Tr ~A:
…3:60†
(2) The product of the eigenvalues equals the determinant of the matrix:
1 2 n ˆ det ~A:
…3:61†
125