Let {\displaystyle \mathbf {x} } be an arbitrary real m -dimensional column vector of {\displaystyle A} such that {\displaystyle |\mathbf {x} |=|\alpha |} for a scalar α . If the algorithm is implemented using floating-point arithmetic, then α should get the opposite sign as the k -th coordinate of {\displaystyle \mathbf {x} }, where {\displaystyle x_{k}} is to be the pivot coordinate after which all entries are 0 in matrix A 's final upper triangular form, to avoid loss of significance.