Introduction to BLAS
Basic Linear Algebra Subprograms or BLAS is a specification created in the 70s/80s by members of academia from US public institutions. The aim was the standardization and speed improvement for low-level linear algebra operations. This initially involved vector operations (Level 1). Over time, BLAS came to support more complex algorithms like vector-matrix operations (Level 2) and in the end matrix-matrix operations (Level 3).
For a general overview, the Netlib Quick Reference Guide is a good start. Be sure to check out the References section of the Quick Reference Guide for more detailed explanations and examples. It cites three research papers that were the basis of this specification. Jack Dongarra seems to be a key figure in this field. You can find those papers in the public domain.
Take a look at what the specification says about matrix operations. Specific operations involving symmetric, hermitian, and triangular matrices exist because these special cases can improve performance. We will focus on GEMM the general matrix-matrix operation:
