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?
"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."

To give one of many examples, the $LU$ decomposition will


represent breaking a matrix into two new matrices, namely $L$ and $U$.  These matrices are
created after some operation on (the) original matrix, the operation being Gaussian elimination.  Then
if used together, that is, the matrices $L$ and $U$ are used together, and
passed through some operation, the operation being matrix multiplication, they
reproduce the original matrix, because $LU=A$.


I think this fits perfectly into what you have said.  As far as I know there is no other commonly used term for this kind of thing.
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.