= 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 minimizing
$$
Q_W(\lambda)=(\mathbf y-\mathbf a-\lambda\mathbf d)^TW(\mathbf y-\mathbf a-\lambda\mathbf d).
$$
For $c>0$, $Q_{cW}(\lambda)=cQ_W(\lambda)$. 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 $0\leq\lambda\leq1$ is imposed in both fits.
For distinct endpoints, the unconstrained formula makes the cancellation visible:
$$
\widehat\lambda_W=\frac{\mathbf d^TW(\mathbf y-\mathbf a)}{\mathbf d^TW\mathbf d},\qquad
\widehat\lambda_{cW}=\widehat\lambda_W.
$$
The differential in <sensitivity of a tie-line fit to its weight matrix> agrees: for a perturbation $\delta W=\varepsilon W$, its numerator is $\varepsilon\mathbf d^TW\mathbf r=0$ 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 $W=\operatorname{diag}(1,4)$. The remaining composition coordinate is fixed by closure. These values give
$$
\mathbf d^TW\mathbf d=0.8,\qquad
\mathbf d^TW(\mathbf y-\mathbf a)=0.508,\qquad
\widehat\lambda=0.635.
$$
Replacing $W$ 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> $1/(\mathbf d^TW\mathbf d)$ then decreases by a factor $c$. The <measurement model> and covariance interpretation must be stated.<Guide to the expression of uncertainty in measurement by JCGM>[Sections 5.1.3 and 5.2.2, model sensitivities and covariance propagation; tie-line result derived here]{ref}
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