While using Least Squares algorithm, I'm trying to get $$f(a_1,a_2,v_1,v_2)=Aa_1+Ba_2+Cv_1+Dv_2-E$$ approximation under the constraint: $$v_1^2+v_2^2=s^2 (s\in\Bbb{R})$$
In the end, I got the following equation with the Lagrange Multiplier: $$\left(M+ \begin{pmatrix} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&\lambda&0\\ 0&0&0&\lambda\\ \end{pmatrix}\right) \begin{pmatrix} a_1 \\ a_2 \\ v_1 \\ v_2 \end{pmatrix}=b$$ $$(M\in\Bbb{R}^{4\times4}, b\in\Bbb{R}^4) $$ And $M$ might not be invertible.
I am trying to get rid of $\lambda$ without any success...
I know that this question has been answered before, but I couldn't understand any of the answers there.
Can someone please send me an article or answer me in a more simplified way?