SEM, this gradient is $$\boldsymbol{g} = \dot{F}(\boldsymbol{\theta})  = \boldsymbol{\Delta}^T \mathbf{V} \left[\mathbf{s} - \boldsymbol{\sigma}(\boldsymbol{\theta})\right],$$ where $\boldsymbol{\Delta}$ is the Jacobian of the implied (co)variances with respect to the parameters, $\boldsymbol{\Delta} = \dot{\boldsymbol{\sigma}}(\boldsymbol{\theta})$. This Jacobian has $p^*$ (the number of unique variances and covariances) rows and $p$ (number of parameters) columns; note that the degrees of freedom of the model equals $p^* - p$. Drawing a parallel with linear regression, the matrix $\boldsymbol{\Delta}$ can be thought of as a design matrix in the linearized mapping of parameters into (co)variances [@savalei_understanding_2014]. Since $\boldsymbol{S}$, $\boldsymbol{\Sigma}$, and $\mathbf{V}$ are positive-definite by assumption, linear dependence in the gradients can only occur as a consequence of a rank deficiency in the Jacobian $\boldsymbol{\Delta}$. Thus, nonconvergence is a direct consequence of rank deficiency of $\boldsymbol{\Delta}$, i.e. when the column rank $\text{rk}(\boldsymbol{\Delta}) < p$.

So-called “inadmissible” estimates also result from rank deficiencies in the Jacobian $\boldsymbol{\Delta}$. “Inadmissible” estimates – such as negative estimates for variance parameters or non-positive definite latent variable corvariance matrices – are possible because the optimization space is usually taken as $\mathbb{R}^p$, which includes “inadmissible” subsets. In fact, for many standard CFA models, *most* of the optimization space is “inadmissible” [see @mulder_equality_2010 for the closely related concept 