# Chi-Square Tests with MLE in Confirmatory Factor Analysis

I have a basic question about confirmatory factor analysis. When performing CFA, Chi-square test is often used. I read from the book that after using MLE for factor extraction (computing covariance matrix), the test statistic $$v=(N-1)F_{ml}$$ is distributed as a chi-squared distribution. For example, on Pg. 63 of Statistical Analysis of Management Data by Hubert Gatignon:

Based on large-sample distribution theory, $$v=(N-1)\hat{F}$$ (where $$N$$ is the sample size used to generate the covariance matrix of the observed variables and $$\hat{F}$$ is the minimum value of the expression $$F$$ as defined by Equation 4.15) is distributed as a chi-squared with number of degrees of freedom...

However, I only know by Pearson's theory, the test statistic $$T=\sum_{j=1}^r \frac{(v_j-\mathbb{E}v_j)^2}{\mathbb{E}v_j}\rightarrow\chi^2_{r-1}$$ where $$v_j$$ is the observation in category $$j$$. The test statistic is clearly different from $$v=(N-1)\hat{F}$$. Could someone explain why $$v$$ can be interpreted as a test statistic distributed as $$\chi^2$$? I searched online but couldn't find much explanation.

In addition, I understand we are trying to test the covariance matrix, but I fail to see how it is connected to $$v$$ and it would be great if someone can explain this too!