the growth factor

When you run elimination, the numbers inside the matrix change at every step, and sometimes they get bigger than anything you started with. The growth factor is the single number that measures how much: it compares the largest-magnitude entry that ever appears anywhere during the elimination to the largest entry of the original matrix. If entries stay roughly the same size, growth is small and you are safe; if they balloon, you are in trouble.

Precisely, the growth factor rho is the ratio of the maximum absolute value of any entry occurring across all the intermediate matrices of the elimination to the maximum absolute value in A. It matters because backward error analysis shows the rounding errors of Gaussian elimination are bounded by something proportional to rho times the unit roundoff: the computed L and U solve a system A + E with the perturbation E small relative to rho. So small growth means backward stability and a trustworthy answer; large growth means the elimination has internally created huge cancellations that swamp the data. Pivoting exists largely to keep rho under control: with partial pivoting every multiplier has magnitude at most 1, which caps rho at 2^(n-1) in the worst case but keeps it tiny (a handful, not millions) for essentially all real matrices.

The growth factor is the honest reason 'pivot for stability', not just 'pivot to dodge a zero', is the right slogan. With no pivoting rho can be enormous even when A is well-conditioned, so the answer is ruined by the algorithm, not the problem. With complete pivoting rho is provably bounded polynomially. The subtle and famous fact is that partial pivoting's 2^(n-1) bound is essentially never approached in practice — pathological growth matrices are vanishingly rare — which is exactly why partial pivoting is trusted everywhere despite lacking an airtight worst-case guarantee.