When doing cryptanalysis on RSA or something number theory based, there are various attacks that all seem to involve number theory (Wikipedia: RSA_(algorithm) ~ Attacks_against_plain_RSA). Similarly, attacks on ECC seem to involve something from ECC (Elliptic curve cryptography related key attacks).
What I'm wondering about is matrix-based (perhaps someone can correct me to the proper term for this) cryptography and cryptanalysis. I'm especially wondering if I'm just stating the obvious by associating the attacks with their original crypto-systems or if there are methods that work for different kinds of systems based on different math theory (an example of this would really be helpful). Particularly the matrix-based (which seems to involve multiple points orthogonal in $R^n$).
Can cryptanalysis based on that be used on other crypto-systems?