Given: A Square Matrix where each entry is an integer (either positive or negative). The magnitude of each integer entry is at most $m$ bits. The size of the Square matrix is $n \times n$.
Objective: Reduce the Square Matrix to a Diagonal Matrix form using Gaussian Row Elimination.
Query: What is the most efficient algorithm and its time complexity for the above?
Comment: Most sources discuss the number of operations but not the total computational complexity. The obvious method is to use the text book method of row elimination. But I am assuming in the worst case, it is possible for the intermediate matrix entries to blow up exponentially in their size (w.r.t. $m$). Thus, I am not clear.
Can someone help with the most efficient algorithm to achieve the above and its time complexity? (a clear reference to a textbook/source where that algorithm as well as its complexity is derived would be equally great).
P.S. I was suggested this link Link but it is not very clear (either Algorithm or its complexity)