Diagonal Matrix Sequences and their Spectral Symbols

The spectral symbols are useful tools to analyse the eigenvalue distribution when dealing with high dimensional linear systems. Given a matrix sequence with an asymptotic symbol, the last one depends only on the spectra of the individual matrices, seen as a not ordered set. We can then focus only on diagonal sequences and sort the eigenvalues so that they become an approximation of the symbol sampling. We show that this is linked to the concept of diagonal Generalized Locally Toeplitz (GLT) sequences, and in particular we prove that any diagonal sequence with a real valued symbol can be permuted in order to obtain a diagonal GLT sequence with the same symbol.