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For real random vectors $\mathbf{a}\in {\mathbb{R}}^p$ and $\mathbf{b}\in {\mathbb{R}}^q$ with finite second moments, their cross-covariance matrix collects the covariances between the two sets of coordinates:It has size $p\times q$, and exchanging the two vectors transposes it: $C_{ba}=C_{ab}^T$. Unlike a covariance matrix of a vector with itself, a cross-covariance matrix need not be square or symmetric.

For an original construction, let centered independent random variables $X,Z$ have variances1and4, and putLinearity of covariance givesThe second entry contains a negative contribution from $Z$, although the total is positive. Also $\mathrm{Var}(b)=13$ andThis joint covariance matrix is positive semidefinite. That restriction couples its diagonal and off-diagonal blocks; an arbitrary rectangular matrix cannot be combined with arbitrary within-vector covariances to produce a valid joint model.

For fixed real matrices $A,B$ and constant vectors $c,d$, direct substitution into the definition givesThis identity propagates shared input fluctuations through two different linear readouts, for example when assessing shared calibration error.

From $n$ independent identically distributed paired observations, the usual unbiased estimate isThe pairing must be preserved: independently shuffling the second set changes the estimated association. NIST's multivariate tutorial gives the coordinate sample-covariance formula and its assembly into a covariance matrix; the rectangular formula above is the block joining the two chosen coordinate groups.^[ref]

Tie-line endpoint covariance records how the measurement errors of the two measured liquid-liquid tie line endpoints vary together. Using independent mole fraction coordinates, represent the two endpoint estimates by column vectors $\mathbf{a},\mathbf{b}$. Their cross-covariance matrix isAn individual off-diagonal block need not be symmetric; the full joint covariance matrix must be a symmetric positive semidefinite matrix. The uncertainties of the two endpoint vectors alone do not determine their cross-covariance matrix.

Near an exactly balanced, interior tie-line phase-fraction fit, let $\mathbf{g}$ be the row-vector sensitivity coefficient defined in tie-line phase-fraction sensitivity. If the overall composition measurement $\mathbf{y}$ is independent of both endpoints, first-order uncertainty propagation givesThe cross term has a positive coefficient because the two endpoint perturbations enter the phase fraction with the same sign. Correlated endpoints can therefore increase the result. The general rule is to propagate the full joint covariance matrix, rather than assume that correlation always increases or always reduces uncertainty.^[ref]

For a stipulated binary mixture example, use $a=0.20$, $b=0.80$, $z=0.44$, so $\lambda =0.40$. Treat $z$ as exact and give each endpoint standard uncertainty $s=0.006$ with correlation coefficient $\rho$. The lever rule yieldsThis gives $0.002$ for $\rho =-1$, about $0.00721$ for $\rho =0$, and $0.010$ for $\rho =1$. These are first-order standard uncertainties under the stipulated joint error model, not bounds on all possible errors.

A shared calibration error affecting the overall composition as well changes the calculation. Suppose the same additive error $c$ affects $a,b,z$ identically. ThenThe shared offset cancels exactly. More generally, a common affine map $x\mapsto mx+c$ with $m\ne 0$ also cancels in this binary ratio. This common affine calibration cancellation in the lever rule concerns only a truly common contribution; separate sample errors remain. If the shared offset has standard uncertainty $0.006$, incorrectly treating its three appearances as independent would assign a spurious contribution $0.01\sqrt{1+0.6^2+0.4^2}\approx 0.01233$ instead of zero.

In general, stack the inputs as $\mathbf{x}=(\mathbf{y},\mathbf{a},\mathbf{b})$ with joint covariance matrix $C$. Let $J$ be the corresponding Jacobian matrix row from tie-line phase-fraction sensitivity, including its residual correction when needed. Then $u^2(\lambda )\approx JC{J}^T$ automatically includes correlations with the overall composition. The measurement guide specifically identifies common instruments, measurement standards and reference data as sources of such correlations.^[ref]

An instructive measurement design analyzes the two equilibrium phases and the overall mixture using one instrument calibration, then repeats the entire procedure with independent instrument calibrations. The within-run contrasts and between-run changes probe different error contributions. Repeated readings under one unchanged instrument calibration cannot by themselves establish the uncertainty of its shared offset. The measurement model must identify which influences are common before the joint covariance matrix can represent them.