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I search the non trivial solution for the system $\bf{Ax}=\bf{b} = \bf{0}$, where the equations are weighted with the matrix $\bf{W}$.

I found 2 different approaches for each part, which I would like to combine, but I am not sure how to do this.

  1. Solving the homogeneous system: Calculate the generalized Eigenmatrix of $\bf{V} = \bf{A^TA}$, where the vector corresponding to the smallest eigenvalue is a solution.

  2. Using classical LSE including weighting: $\bf{x} = \bf{(A^TWA)^{-1}A^TWb}$

I think another option is to change the problem to LSE with constraint ($\bf{||x||} = 1$} for example). My problem with that is, that I need to implement this efficiently on an microcontroller. So far I couldn't find a comprehensible algorithm for that (certainly due to my limited knowledge). The advantage of 1. is the easy implementation.

So finally, the questions:

A) Is there a way to combine approach 1 and 2?
B) If not, what is the solution path for LSE with constraints?

user6522399
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1 Answers1

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I found the answer with an intermediate step posted here by Matt L.

Following his explanation, the weighted error $e$ is:

$e = \bf{W(Ax-b)}$

leading to the system of equations:

$\bf{A^TW^TWAx}= \bf{A^TW^TWb}$

with $\bf{b}=\bf{0}%$ we have:

$\bf{A^TW^TWAx}= \bf{0}$

to which we can apply the solution 1) from the question, calculating the generalized eigenmatrix from V with:

$V = \bf{A^TW^TWA}$

The eigenvector corresponding to the smallest eigenvalue is now the solution.

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