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Eigenvalue Analysis 165
In particular, when α = 0, the real part of these eigenvalues is constantly equal to 1
2 .
The proof of this fact is quite simple and is left as exercise.
Finally, we consider problem (7.4.28) in which the eigenvalues and eigenfunctions
are respectively denoted by λn,m and pn,m, 1 ≤ m ≤ n. This time, the theory is simple.
Comparing (7.4.28) with the differential equation (1.7.1), we get λn,m = 2(m − 1) + µ
and pn,m = Hm−1, 1 ≤ m ≤ n. Therefore, the eigenvalues are real and positive when
µ > 0.
8.3 Condition number
The condition number is a useful quantity to estimate the speed of convergence of an
iterative method for the solution of linear systems.
It is well-known that a real n ×n matrix D corresponds to a linear operator from R n
to R n . Denoting by L(R n ,R n ) the space of linear mappings from R n in R n , we can
define the norm of D (see for instance isaacson and keller(1966), p.37) as follows:
(8.3.1) D L(R n ,R n ) := sup
¯x∈R n
¯x=0
D¯x
¯x
= sup
¯x∈R n
¯x=1
D¯x,
where by · we indicate the canonical norm in R n . The expression in (8.3.1) actually
satisfies all the properties required in section 2.1. In practice, the norm of D measures
the maximal deformation, when applying the matrix to the vectors of the unitary sphere
of R n .
We remark that, for any n × n matrix D and any ¯x ∈ R n , one has the inequality
(8.3.2) D¯x ≤ D L(R n ,R n )¯x.
At this point, we can define the condition number of a nonsingular matrix D. This is
the real positive number given by
(8.3.3) κ(D) := D L(R n ,R n )D −1 L(R n ,R n ),