Since the iteration matrix M defined defined by the canonical eigenvalue problems from Section 1.2 is symmetric, M has a complete set of orthogonal eigenvectors which can be be denoted by
Then,
where, for i>0, 
is the 
Chebyshev iterate
generated by Equation (11), 
is the initial iterate, and 
is an eigenvalue of M
corresponding to the eigenvector 
.
Consider the inner product of the 
and 
iterates, i.e.,
Equation (18) is equivalent to the continuous integral
when 
is defined to be the following discrete non-negative
distribution ([10])
By choosing l=0 in Equation (19) and 
,
it follows that
Note that the final orthogonal polynomial 
has a zero at each
eigenvalue i.e., 
.
Hence, at each step of the Chebyshev semi-iterative method, we can
extract the 
modified moment
The extraction of moments from iterates can be accelerated by using the recurrence relations for the Chebyshev polynomials defined in Equation (9). Specifically, one can show (see [22]) that
Note that the polynomial 
in Equation (15)
associated with the modified moments 
and 
satisfies
where the coefficients 
of the polynomials 
in Equation (16) are given by
