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I am trying to understand the Kalman filter, and I am struggling to find out how to choose the transition matrix (denoted as $\mathbf F(k)$ or $\mathbf A(k)$. This matrix is used to update the weight error covariance matrix as follows

$$\mathbf P(k+1) = \mathbf F(k+1) \mathbf P(k) \mathbf F^\mathrm T (k) + \mathbf Q$$

In the paper On the Intrinsic Relationship Between the Least Mean Square and Kalman Filters it is description of "The Kalman filter for deterministic states.", what does not use the part with prediction, so it is without the transtion matrix.

But there is also the general Kalman filter, with the prediction according to transtion matrix. I have not understand how this matrix should be obtained.

I quess, that this transition matrix is from state space formulation of the model. But what should I do if I do not have a model? Is this mean that the general Kalman filter is usable only If I have exact model of the system?

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Simple: if you do not have a model, you cannot apply the Kalman filter. Or you could and make up a model, but you cannot expect any of the optimality properties of the filter to hold. Based on that paper, it seems they assume that $\mathbf{x}_k$ is known or at least directly measureable...

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