As discussed by Paige and Saunders in [168] and by Golub and Van Loan in [109], it is straightforward to derive the conjugate gradient method for solving symmetric positive definite linear systems from the Lanczos algorithm for solving symmetric eigensystems and vice versa. As an example, let us consider how one can derive the Lanczos process for symmetric eigensystems from the (unpreconditioned) conjugate gradient method.
Suppose we define the 
matrix 
by
and the 
upper bidiagonal matrix 
by
where the sequences 
and 
are defined by the standard
conjugate gradient algorithm discussed in §
.
From the equations
and 
, we have 
, where
Assuming the elements of the sequence 
are 
-conjugate,
it follows that
is a tridiagonal matrix since
Since span{
} =
span{
} and since the elements of
are mutually orthogonal, it can be shown that the columns of
matrix 
form an orthonormal basis
for the subspace 
, where
is a diagonal matrix whose 
th diagonal element is 
. The columns of the matrix 
are the Lanczos vectors (see
Parlett [171]) whose associated projection of is
the tridiagonal matrix
The extremal eigenvalues of 
approximate those of the
matrix . Hence,
the diagonal and subdiagonal elements of in
(), which are readily available during iterations of the
conjugate gradient algorithm (§),
can be used to construct after 
CG iterations. This
allows us to obtain good approximations to the extremal eigenvalues
(and hence the condition number) of the matrix while we are generating
approximations, 
, to the solution of the linear system 
.
For a nonsymmetric matrix , an equivalent nonsymmetric Lanczos
algorithm (see Lanczos [142]) would produce a
nonsymmetric matrix in () whose extremal eigenvalues
(which may include complex-conjugate pairs) approximate those of .
The nonsymmetric Lanczos method is equivalent to the BiCG method
discussed in §.
