Storage of matrices for experiments in SQL database

Question

I need to store experimental matrices in a database. I started by compiling an ERD and ran into a problem. The tables "Matrix" and "Matrix_element" must be required (this is a requirement). The problem is that the matrices for experiments have string names, so I would like to have access by name, and not by id. At the same time, it seems not correct to replace "matrix_id" with a string type, because the names can be quite long (something like "TempNreg410"), while the matrix size can be 5000x60000, and there are hundreds of such matrices. It turns out the repetition of a long line a huge number of times. Please advise how to proceed. Below are 2 diagrams that I have been able to come up with so far.