Scalar rescaling invariance of a tie-line weight matrix

Scalar rescaling invariance of a tie-line weight matrix means that multiplying every fit weight by the same positive number leaves the fitted phase fraction unchanged. Let endpoint compositions $\mathbf{a},\mathbf{b}$ and overall composition $\mathbf{y}$ be expressed in independent mole fraction coordinates. With $\mathbf{d}=\mathbf{b}-\mathbf{a}$, a fixed symmetric positive-definite matrix $W$ defines the tie-line phase-fraction fit by minimizingFor $c>0$, $Q_{cW}(\lambda )=c{Q}_W(\lambda )$. Multiplication by a positive constant preserves every comparison between objective values, so the sets of minimizers agree. The proof also applies when the same constraint $0\le \lambda \le 1$ is imposed in both fits.

For distinct endpoints, the unconstrained formula makes the cancellation visible:The differential in sensitivity of a tie-line fit to its weight matrix agrees: for a perturbation $\delta W=\epsilon W$, its numerator is $\epsilon {\mathbf{d}}^{T}W\mathbf{r}=0$ by the fit's normal equation. Thus the scalar direction is insensitive even when the residual is nonzero.

In an original example, take $\mathbf{a}=(0.65,0.10)$, $\mathbf{b}=(0.25,0.50)$, $\mathbf{y}=(0.42,0.36)$ and $W=\mathrm{diag}(1,4)$. The remaining composition coordinate is fixed by closure. These values giveReplacing $W$ by $7W$ gives the same fraction, although the minimum objective increases from0.00072to0.00504. Changing only the second diagonal weight from4to7 instead changes the fitted fraction to0.640625. Relative weights determine the geometry; their common multiplier does not.

Point-estimate invariance does not by itself establish invariance of physical measurement uncertainty. Merely rewriting the same objective with $cW$ leaves propagation through the same measured-input covariance matrix unchanged. If instead $W={\Sigma }^{-1}$ represents absolute precision and $cW$ asserts a genuinely different covariance $\Sigma /c$, that is a different noise model. With exact endpoints, the linear-model variance $1/({\mathbf{d}}^{T}W\mathbf{d})$ then decreases by a factor $c$. The measurement model and covariance interpretation must be stated.^[ref]