First introduce some additional notations. Let the dimension of

\(\mathbf {F}(\varvec{\theta })\) be

\(n_r \times n_z\), the rank of

\(\mathbf {F}(\varvec{\theta })\) be

\(n_f\), and the length of

\(\varvec{\theta }\) be

\(n_{\varvec{\theta }}\). For the equation

$$\begin{aligned} \mathbf {r}= \mathbf {F}(\varvec{\theta }) \mathbf {r}_z \end{aligned}$$

(8.100)

to be compatible, it must hold that, cf (

8.57)

$$\begin{aligned} n_r \ge n_{\varvec{\theta }} + n_z \; . \end{aligned}$$

(8.101)

The case to consider and discuss is when

\(\mathbf {F}\) is rank-deficient, that is when

$$\begin{aligned} n_f < n_z \; . \end{aligned}$$

(8.102)

Assume that

\(\mathbf {F}\) can be factorized as

$$\begin{aligned} \mathbf {F}(\varvec{\theta }) = \mathbf {F}_1(\varvec{\theta }) \mathbf {F}_2(\varvec{\theta }) \; , \end{aligned}$$

(8.103)

where

\(\mathbf {F}_1(\varvec{\theta })\) is

\(n_r \times n_f\) and

\(\mathbf {F}_2(\varvec{\theta })\) is

\(n_f \times n_z\). As

\(\mathrm{rank} (\mathbf {F}) = n_f\), it must then hold that

\(\mathbf {F}_1(\varvec{\theta })\) has full column rank equal to

\(n_f\). Equation (

8.100) can now be transformed into

$$\begin{aligned} \mathbf {r}= & {} \mathbf {F}_1(\varvec{\theta }) \mathbf {g}\; , \end{aligned}$$

(8.104)

$$\begin{aligned} \mathbf {g}= & {} \mathbf {F}_2(\varvec{\theta }) \mathbf {r}_z \; . \end{aligned}$$

(8.105)

Consider first (

8.104) only, and treat the vector

\(\mathbf {g}\), which has length

\(n_f\), as consisting of auxiliary unknowns. It turns out that this equation is overdetermined, as

$$\begin{aligned} \mathrm{dim}(\mathbf {r}) = n_r \ge n_{\varvec{\theta }} + n_z > n_{\varvec{\theta }} + n_f = \mathrm{dim} (\varvec{\theta }) + \mathrm{dim} (\mathbf {g}) \; . \end{aligned}$$

(8.106)

As

\(\mathbf {F}_1\) has full rank, one can expect that (

8.104) should lead to a sound least squares solution. Once,

\(\varvec{\theta }\) and

\(\mathbf {g}\) are found, (

8.105) can be treated with

\(\mathbf {r}_z\) as unknowns. This system is under-determined, and

\(\mathbf {r}_z\) cannot be determined uniquely. Still, it is

\(\varvec{\theta }\) that is of primary interest to estimate.

It may be of interest to consider also another type of factorization that is closely related to (8.108).