# Optimal pivoting strategy in LU factorization

I'm currently reading the book Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III, working my way through the required lectures for my Numerical Analysis class. The current subject is LU factorization, and while reading Lecture 21 about Pivoting (in Gaussian Elimination), I came across the statement that in practice, partial pivoting is equally as good as complete pivoting, while requiring inspection of a much smaller number of entries.

I'm pretty sure I understand correctly what partial and complete pivoting mean, namely with complete pivoting the entire $$A_{k:m, k:m}$$ submatrix is examined while with partial pivoting only the $$A_{k:m, k}$$ column is examined for an optimal pivot (the largest possible value, that is (right?)).

This seems very wrong to me and I'm surely missing some trivial argument for why this must be true. How can we be sure that part of the $$k$$th column of A always contains the biggest value in the entire submatrix? Am I right in assuming that the author meant to say that the increased gain from choosing the optimal value in the entire submatrix would be vastly overshadowed by the sheer amount of time needed to find it, and that only considering the first column of the submatrix yields good results?