Matrix decomposition definition Wikipedia says "In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems."
But in my opinion decomposition term should be used to represent breaking a matrix in different sub-matrices or some new matrices created after some operation on original matrix which if used together and passed through some algorithm(not necessarily product), shall reproduce the original matrix.
Is there some different terminology to represent what I am expecting to say?  

I have tried answering this question, It was long so I am putting it as link. I am not a mathematician but programmer, please provide some inputs.
