The diagonal form of the matrix is achieved by orthogonal transformation.

thanks for your response. Do you have any worked out example for the same?

Generally, when working on a matrix of unknown dimensions, we statically declare a matrix of sufficient size, then fill according to the size of your dataset

Dear Anuradha,

I'm not sure I understand your question well, but I'll try to give you an appropriate answer. First of all, the term "initial weight matrix" sounds strange for me, because in most applications it remains the same. (There are some exceptions, e.g. when the goal of our team was to decrease the complexity (= the number of concepts) of FCM by a special model reduction method, but it is required only if the model is over-sized.)

If you have a series of historical data, you already know the number of concepts. For example, if the historical data is generated by a set of sensors, the number of concepts is usually equal to the number of sensors. There are several possible methods for the construction of the weight matrix. One class of these algorithms are based on the Hebbian learning. For example, Elpiniki Papageorgiou developed some of these: "Active Hebbian Learning", "Fuzzy Cognitive Map Learning Based on Nonlinear Hebbian Rule", etc., but several other well-known researchers worked on similar techniques, too. The other class of algorithms is based on evolutionary algorithms. I have personal experiences only with these. I developed a Bacterial Evolutionary Algorithm that starts with a collection of random-generated weight matrices, and continuously improves them. The objective function includes a complete FCM simulation and I compare the squared sum of differences between the values of the original time series data and the other data generated by the FCM simulation using the current version of the weight matrix.  Of course, my goal is to reduce this difference (error) by modifying the matrices until I reach a predefined limit. There are several minor technical problems to solve (e.g. what to do if different number of time steps are needed in the two time series or how to handle the case when you have a finite length time series but the generated model behaves chaotically, etc.) but the essence of the technique is this.