Are there any differences between them (w.r.t) the calculation. I know the sample covariance matrix uses the sample data and the population covariance matrix uses the random varibles.
Solved – Are there any differences between the sample covariance matrix and the population covariance matrix
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Covariance matrix = Variance-covariance matrix
I'm not sure I understand all of your notation.
You have three latent variables, so why does the covariance matrix of latent variables have 9 rows and columns? (And what is $\gamma$?) (And what are $\phi$ and $\theta$ - different authors use symbols differently, so it's best to define what you mean).
You have $S$, your sample covariance matrix.
You have a model implied covariance matrix $\Sigma$ (or sometimes $\Sigma(\theta)$.
You have a model - the model implies $\Sigma(\theta)$.
In a CFA model (such as you have):
$\Sigma(\theta) = \Lambda\Phi\Lambda' + \delta$
Where $\Lambda$ is the matrix of loadings, $\delta$ is the matrix of errors (E in your case, which is a diagonal matrix), and $\Phi$ is the covariances of the latent variables (F). If you make the variances equal to 1, and the covariances equal to zero, then $\Phi$ is an identity matrix and you can ignore it.
If you don't want to think about it in terms of matrix algebra, the implied covariance between two items is the product of the paths between them. So the covariance of $V1$ and $V2$ is $\lambda_1 \times \lambda_2$ and the covariance of $V1$ and $V4$ is $\lambda_1 \times cov(F1, F2) \times \lambda_2$. (I'm assuming that all latent variable variances are equal to 1.00, because that makes things easier.) This is what the equation showing $\Sigma(\gamma)$ is doing - but it's rather hard to see without looking at the path diagram at the same time.
So you find the values for the unknowns in the model to try to ensure that $\Sigma(\theta)$ and $S$ are as similar as possible. There are varies ways to measure similarity, but the most common is maximum likelihood (ML).
Find the distance $F_{ML}$
$F_{ML} = log(\Sigma(\theta)) + tr(S\Sigma(\theta)^{-1}) - log(\Sigma) - p$
Where $p$ is the number of variables in the model. If $\Sigma(\theta) = S$ then $F_{ML} = 0$.
You can calculate the $\chi^2$ statistic using:
$\chi^2 = F_{ML} \times (N - 1)$ where $N$ is the sample size. (Note that $N-1$ is used by legacy software such as LISREL, EQS, and Amos, which date back to the use of Wishart likelihood for sampling variability of covariance matrices; more recent software like Mplus and lavaan use normal likelihood and multiply $F_{ML}$ by $N$.)
Most of the other fit indices are derived in some way from these values (obvious exceptions are the GFI, which no one uses any more, and SRMR, which is based on the difference between $S$ and $\Sigma$).
You can play around with this using whatever program you like that can do matrix algebra and has an iterative solver. Quite some time ago, I wrote a paper showing how to do this with MS Excel, which you can find here: https://link.springer.com/content/pdf/10.3758/BF03192739.pdf. It's been a while since I used Excel, so I dont' know if this still works, but here's the sheet.
Best Answer
Sample covariance matrix is an estimation for the population covariance matrix. As all estimators, it uses sample data and is experimental. On the other hand, the population statistics is theoretical and can be calculated when you know the joint distribution.