How to interpret the endogenous factor matrix in the General Structural Equation Model? Bollen (1989) introduces a general structural equation model of the (matrix) form:
$$\eta = \beta \eta + \gamma \xi + \zeta$$
$$y = \lambda \eta + \epsilon$$
Most textbooks (e.g. Depaili 2021) presenting the general model describe $\beta$ is a $m \times m$ matrix of coefficients "relating the $m$ endogenous latent factors." However, the Bollen text and others don't describe this matrix in much detail. Accordingly, I'm hoping someone can help explain the contents and dimensionality of the matrix.
Hopefully, an answer would hit on the following points:

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*Why the matrix is square and not lower-triangle since presumably we don't "regress variables on themselves."

*Why the matrix is not symmetric. Is this because these are directed paths (regressions) rather than covariances?

*Are the elements of this matrix the marginal, bivariate correlations between each factor? Or are they the partial correlations? For example, suppose $m > 2$ (more than 2 endogenous latent factors). How would we describe $\beta[1,2]$ or $\beta[2,1]$?


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*I think it is lower triangular (if the model has no feedback loops). See, for example: https://web.pdx.edu/~newsomj/semclass/ho_lisrel%20notation.pdf

*Yes, they're regressions.

*They're regression parameters, the same as if we were doing multiple regression.

It's less common for people to worry about the distinction between exogenous factors and endogenous factors - this was an issue when computers had less memory and were slower, but most programs don't care, so there's no need to distinguish the beta and gamma matrices.
