Let T be as in the preceeding section, and let 
be a solution to
We consider the inverse problem of recovering the diagonal matrix Q
in 
from
Again we put 
with L from Theorem 2.1. We obtain
With 
the recursion can be written as
We claim that for 
and 
where
We prove (5.3) by induction with respect to k.
The case k = 0 is obvious.
Assume (5.3) to be correct up to some k with
. From (5.2) we get
Since S is tridiagonal, only components 1 through n-k of 
enter 
for i < n - k. Hence, by the induction
hypothesis,
This is (5.3) with k replaced by k+1. Thus (5.3) is established.
We use (5.3) for i = 1 only, yielding for 
Introducing the row vectors
we have
and, since S, 
commute,
Thus 
is the analogue to 
with T replaced by 
:
This shows that the matrix 
is upper
triangular with diagonal elements
It follows that the systems
where 
are uniquely solvable for 
provided that 
, 
.
Given that 
, we can determine the matrix
from our data. The matrix
is upper triangular with diagonal elements
Thus computing L, U amounts to doing an LU-Decomposition on the matrix Z,
with the diagonal of U being known. Once L, U are known, Q is computed very much in the same fashion as in the preceding section.
