Summary: Lecture 16: Some remarks on pivot columns of a matrix
Definition: Let A be an (n × m) matrix with columns A = [v1, . . . , vm]. A
column vk, 1 k m, is called a pivot column if there is a sequence of elementary
matrices E1, . . . , EN such that
Ae = EA = E1 · · · EN A
is in echelon form and the k-th column of Ae contains a pivot element.
The following practical question arises with this definition: There are many dif-
ferent echelon forms for a matrix. How do we decide whether some column is a
pivot column ? Assume Ae = EA is an echelon form for A. If the k-th column of
Ae contains a pivot element then vk is a pivot column. But what if this is not the
case ? There may be another echelon form Ae = E A where the k-th column has a
pivot element. But how do we find the 'right' echelon form ? The answer is that
we do not need to. Any echelon form does the job.
Proposition: Let A = (aij) and B = (bij) be matrices in echelon form such that
B can be transformed into A by row reduction, i.e. there are elementary matrices
E1, . . . , EN such that A = E1 · · · EN B.
If (ai1, . . . , aim) and (bi1, . . . , bim) are two nonzero rows then they have their pivot
elements in the same column, i.e.
ji = li