or purposes of financial
problems we are always interested in equations of the
form
where the
is some differential operator acting on some spacial variables. Finite
difference scheme for such equation regularly has the
form


(Evolution equation)

where the
is some spacial finite difference operator with complete system of
eigenvectors for every
.
Hence, we start from some initial condition
and use the evolution equation (
Evolution
equation
) to recursively evaluate
for all times. Let
is the spectrum of
.
The
condition
is a sufficient condition for stability of the scheme in such a situation.
Two more statements are directly relevant to the subject of stability.
Proposition
(Minimax theorem). Suppose
is a bounded normal
(
)
operator in a Hilbert space.
Then


(Minmax theorem)

The spectral minimax theorem is a consequence of the following decomposition
statement.
Observe that the above statement provide a complete recipe for determining
norm of a normal matrix and thus establishing stability of an evolution scheme
with a normal operator. Naturally, we would like to have some extension to the
situation without normality. The following considerations show that for most
practical purposes there is no such extension. Consequently, one uses energy
considerations and Galerkin (finite element) technique (see the chapter
(
Finite elements chapter
)) if the
normality cannot be preserved.
We conclude from the above theorem that
Consequently, the matrix is normal if and only if all
are equal to 1.
To see that the spectral analysis does not deliver simple stability results in
the situation without normality let us evaluate the norm of the matrix
.
In
general,
