3.1 Systems of Equations
109
Power Bounds on Matrices
The preceding stability conditions may be expressed in terms of the amplification
matrix after introducing the concept of matrix norms. The norm of a matrix is
defined in terms of the more familiar vector norm such that if B is an M x N
matrix and z a column vector of length N, then
IIBII = sup IIBzlI = sup 11Bz11
IIzll= 1
IIzll;eO IIzli
In particular, if bij is the element in the ith row and jth column of B, then
N
IIBlloo = max L Ibijl
l :-;:i:-;:M j = 1
and
where B* is the conjugate transpose of B, and p is the spectral radius, defined as
the maximum in absolute value of the eigenvalues of a square matrix.
Necessary and sufficient conditions for the stability of a constant-coefficient
linear system may be expressed using this norm notation as
IIAkli s I
(3.3)
for nongrowing solutions, and as
(3.4)
in cases where the true solution grows with time or where the interest is only in
ensuring that the numerical solution will be sufficiently stable to converge in the
limit of Ax, ßt
O. (Once again, CT depends on time, but not on S» and ßt.)
Up to this point, the stability analysis for the single scalar equation and the
system are essentially the same. The difference between the two arises when one
attempts to reduce the preceding conditions on IIAk 11 to a constraint on IIAkll.
In the scalar case the necessary and sufficient condition that IAk 1 I is just
1Ak 1 S 1. On the other hand, when the amplification factor is a matrix, the necessary and sufficient conditions for Ak to be "power bounded" are rather complex.
Since IIAk 11
tion IIAk 11
IIAk II
n (by the fundamental properties of any norm), the condiI will ensure stability. This condition is not, however, necessary for
stability, as may be seen by considering the matrix
( 1-1)
0
E =
-1
'
for which 11Eil 00 = 2, IIEII2 = (3 + 0)/2, and yet for all positive integers m,
E
2m
is the identity matrix, whose norm is unity.