The name is absent

17

We shall show that Ьц ≤ ∣M∣ implies ∖b_ji∖ < b_jj. System ( 4.1) also implies that
∏+1

ɪɪ) λ_ib_ih = 0,h = 1,∙ ∙ ∙ ,n.
i=l

It is easily derived that

a,∙m ≥ a_j∙ ∣m ∣

and

MIMI — λjbjj.

Notice that at least one of the above inequalities holds with strict inequality, say,
λiba > Aj ∖bi_j ∣. Moreover, it holds that A₈M ≤ Λ√∣6√₇-1. All of this together implies
that

Aj∣M∣ < A_iM ≤ a⅛_j∣ ≤ AjM.

It follows that

∣M∣ < Mj-

We are now ready to prove that the new generated column j, denoted by
(M', ’ ’ ’ 5 M-+∣)√ M∙ preserves the same sign pattern as before. Note that

Mj = Mj + сМг, A = 1,   , И + 1.

It is readily seen that

M = — d ≤ 0 and ∣M∣ < M'

Mj ≤ θ for all /г, h M M j∙

Observe that

AiM ≤ А_г|М1
= А_г(сМ + d)
— M Mt
= M(MjA^cIMM)-

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