Aligning matrices, normalization. Calculating coefficients.

Question

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any solutions.

Answer 1

Least squares via normal equations

The linear system is $$ \begin{align} \mathbb{A} c & = \mathbb{B} \\ % \left( \begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \\ \end{array} \right) % \left( \begin{array}{cc} b \\ a \\ \end{array} \right) % &= % \left( \begin{array}{cc} 3 \\ 4 \\ 5 \\ 6 \\ \end{array} \right) % \end{align} $$ The normal equations are $$ \begin{align} % \mathbb{A}^{T}\mathbb{A} a &= \mathbb{A}^{T}\mathbb{B} \\[2pt] \left( \begin{array}{cc} 4 & 10 \\ 10 & 30 \\ \end{array} \right) % \left( \begin{array}{cc} b \\ a \\ \end{array} \right) % &= % \left( \begin{array}{cc} 18 \\ 50 \\ \end{array} \right) % \end{align} % $$ with the solution $$ \boxed{ \left( \begin{array}{cc} b \\ a \\ \end{array} \right)_{LS} = \left( \begin{array}{cc} 2 \\ 1 \\ \end{array} \right) } $$ The residual error vector is $$ r_{LS} = \mathbf{0}, $$ the fit is exact, which implies the data vector $\mathbf{B}$ is in the column space of $\mathbf{A}$. The prescription in terms of column vectors is $\mathbb{B}=2\mathbb{A}_{1} + \mathbb{A}_{2}$.

Least squares via calculus

The target of minimization, the merit function, sums over the rows $r$ and columns $c$: $$ M(a,b) = \sum_{r=1}^{2}\sum_{c=1}^{2} \left( \mathbf{A}_{rc} - a \mathbf{B}_{rc} - b\right)^{2} $$ which leads to $$ \begin{align} % \left( \begin{array}{cc} % \sum^{2}_{r=1}\mathbf{A}_{r,1}\mathbf{A}_{r,1} & \sum^{2}_{r=1}\mathbf{A}_{r,1}\mathbf{A}_{r,2} \\ % \sum^{2}_{r=1}\mathbf{A}_{r,2}\mathbf{A}_{r,1} & \sum^{2}_{r=1}\mathbf{A}_{r,2}\mathbf{A}_{r,2} \\ % \end{array} \right) % % \left( \begin{array}{cc} b \\ a \\ \end{array} \right) % &= % \left( \begin{array}{cc} % \sum^{2}_{r=1}\mathbf{A}_{r,1}\mathbf{B}_{r} \\ % \sum^{2}_{r=1}\mathbf{A}_{r,2}\mathbf{B}_{r} \\ % \end{array} \right) % \end{align} % $$ which yields the same solutions.