 Identifiability from linear observations

Identifiability from linear observations asks which features of an unknown vector $\theta \in {\mathbb{R}}^p$ are fixed by observing $y=A\theta$. Two parameter vectors are observationally equivalent exactly when their difference lies in the kernel of a linear map $A$:Thus the full vector is identifiable precisely when $\ker A=\{0\}$, equivalently when $A$ has full column matrix rank. Repeating the same rows can improve precision in a noisy measurement without increasing this rank.

A linear target $g(\theta )=c^{\mathsf{T}}\theta$ requires less information. It is identifiable if and only if $c$ is in the row space of $A$. If $c=A^{\mathsf{T}}d$, then $g(\theta )=d^{\mathsf{T}}y$, so the target is recovered directly. Conversely, an identifiable target must vanish on every observationally invisible difference: $c^{\mathsf{T}}h=0$ for every $h\in \ker A$. Orthogonality to the kernel is exactly row-space membership. This is a concrete instance of quantity-specific identifiability; the primary identification literature also treats identification of individual parameter components by rank conditions.^[ref]

For an original example, supposeBoth $(1,4,3)$ and $(2,3,4)$ fit perfectly. Their difference is a multiple of $(1,-1,1)$, which spans the kernel. The target ${\theta }_1-{\theta }_3=y_1-y_2=-2$ is identified, whereas ${\theta }_1+{\theta }_2+{\theta }_3$ takes values 8 and 9 and is not. A new independent measurement of ${\theta }_2$ restores full identification; another copy of $y_1$ does not.

The same distinction applies to a statistical model $Y\sim N(A\theta ,\Sigma )$ with known positive-definite covariance matrix $\Sigma$: two parameters give the same observable law precisely when their means agree. A Moore-Penrose pseudoinverse selects one solution when the system is underdetermined, but that numerical convention does not show that the selected vector is the unique physical parameter. Additional restrictions must be justified as assumptions, not silently supplied by a solver.