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Normalization with tex2html_wrap_inline2505


Let tex2html_wrap_inline2637 . As tex2html_wrap_inline2639 , i.e., the sum of squared coefficients, can never be zero for a conic, we can set tex2html_wrap_inline2641 to remove the arbitrary scale factor in the conic equation. The system equation tex2html_wrap_inline2579 becomes


where tex2html_wrap_inline2645 .

Given n points, we have the following vector equation:


where tex2html_wrap_inline2649 . The function to minimize becomes:


where tex2html_wrap_inline2651 is a symmetric matrix. The solution is the eigenvector of tex2html_wrap_inline2653 corresponding to the smallest eigenvalue (see below).

Indeed, any tex2html_wrap_inline2561 symmetric matrix tex2html_wrap_inline2653 (m=6 in our case) can be decomposed as




where tex2html_wrap_inline2661 is the i-th eigenvalue, and tex2html_wrap_inline2665 is the corresponding eigenvector. Without loss of generality, we assume tex2html_wrap_inline2667 . The original problem (4) can now be restated as:

Find tex2html_wrap_inline2669 such that tex2html_wrap_inline2671 is minimized with

tex2html_wrap_inline2673 subject to tex2html_wrap_inline2675 .

After some simple algebra, we have


The problem now becomes to minimize the following unconstrained function:


where tex2html_wrap_inline2677 is the Lagrange multiplier. Setting the derivatives of J with respect to tex2html_wrap_inline2681 through tex2html_wrap_inline2683 and tex2html_wrap_inline2677 yields:


There exist m solutions. The i-th solution is given by


The value of tex2html_wrap_inline2695 corresponding to the i-th solution is


Since tex2html_wrap_inline2667 , 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 tex2html_wrap_inline2653 corresponding to the smallest eigenvalue.

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996