Determine 9 variables by 3 equations with approximation

Question

I have an equation in the form of Q*d=z, where Q is 3by3 matrix of variables, and d and z are vectors of 3 known numbers. What would be the best way to compute all 9 elements of matrix Q, provided that i know the approximate values for all 9 elements?

Answer 1

First off, since your variables are in $Q$, you can consider each row independently . $$q_{i1}d_1+q_{i2}d_2+q_{i3}d_3=z_i.$$ Let's drop the row index $i$ for brevity. $$q_{1}d_1+q_{2}d_2+q_{3}d_3=z.$$ You already have approximate values, let's call them $\hat q_1, \hat q_2, \hat q_3$. If you put them into your equation you get an approximate $\hat z$: $$\hat z = \hat q_{1}d_1+ \hat q_{2}d_2+ \hat q_{3}d_3.$$ Since you want to get $z$ exactly, you have to offset the error $r = z - \hat z$. $$ z = \hat z + r = \hat q_{1}d_1+ \hat q_{2}d_2+ \hat q_{3}d_3 + r.$$ You could distribute this error equally among all 3 variables as: $$q_i = \hat q_i + \frac{1}{3d_i}r.$$ Alternatively, you can try to find the solution that minimizes the squared error $$(q_1-\hat q_1)^2+(q_2-\hat q_2)^2+(q_3-\hat q_3)^2$$.

Answer 2

A good method is Lagrange Multipliers. Though in this case it is a bit complicated.
Suppose $$Q=\left[\begin{array}{ccc}a+\alpha&b+\beta&c+\gamma\\d+\delta&e+\epsilon&f+\zeta\\g+\eta&h+\theta&i+\iota\end{array}\right]$$ Let $d$ be the vector $(p,q,r)$ and $Z=(x,y,z)$.
The Lagrange expression is $$L = \alpha^2+\beta^2+\gamma^2+...+\iota^2+...\\ \lambda(p(a+\alpha)+q(b+\beta)+r(c+\gamma)-x)+...\\ \mu(p(d+\delta)+q(e+\epsilon)+r(f+\zeta)-y)+...\\ \nu(p(g+\eta)+q(h+\theta)+r(i+\iota)-z)$$ $L$ is a function of eleven unknowns $\alpha,...,\iota,\lambda,\mu,\nu$.
Differentiate $L$ with respect to all eleven variables. All eleven derivatives should equal zero. You get eleven linear equations in eleven variables.

Answer 3

It is an ill defined problem and the OP has not much thought about a good presentation. The Michael's solution consists in minimizing the squared error, method also indicated in Andre's post. There are $12 $ unknowns and $12$ relations ($9$ derivatives and the relation $Qd=z$). We can write the equations in compact form (for any dimension $n$) as follows. $Q_0$ is the approximated solution.

We minimize $\phi(Q)=tr((Q-Q_0)^T(Q-Q_0))$ under the linear condition $Qd-z=0$. Then $\nabla\phi(Q)=2(Q-Q_0)$ and the Lagrange's equalities are $Q-Q_0+\begin{pmatrix}\lambda_1 d^T\\\cdots\\\lambda_n d^T\end{pmatrix}=0$.

Finally, the $(\lambda_i)_i$ are defined by $Q_0d-z=\begin{pmatrix}\lambda_1||d||^2\\\cdots\\\lambda_n||d||^2\end{pmatrix}$ and $Q=Q_0-\begin{pmatrix}\lambda_1 d^T\\\cdots\\\lambda_n d^T\end{pmatrix}$.