Let . As
, i.e.,
the sum of squared coefficients, can never be zero for a conic, we can set
to remove the arbitrary scale factor in the conic equation.
The system equation
becomes
where .
Given n points, we have the following vector equation:
where .
The function to minimize becomes:
where is a symmetric matrix. The solution is
the eigenvector of
corresponding to the smallest eigenvalue
(see below).
Indeed, any symmetric matrix
(m=6 in our case) can be
decomposed as
with
where is the i-th eigenvalue, and
is the corresponding
eigenvector. Without loss of generality, we assume
. The
original problem (4) can now be restated as:
Findsuch that
is minimized with
subject to
.
After some simple algebra, we have
The problem now becomes to minimize the following unconstrained function:
where is the Lagrange multiplier. Setting the derivatives of J with
respect to
through
and
yields:
There exist m solutions. The i-th solution is given by
The value of corresponding to the i-th solution is
Since , the first solution is the one we need
(the least-squares solution), i.e.,
Thus the solution to the original problem (4) is the eigenvector of
corresponding to the smallest eigenvalue.