Lemma \[lemmasust\], there exists a pencil $P(s)$ such that $\rank P(s)=1$ and $A(s)+P(s){\ensuremath{\stackrel{s.e.}{\sim}}}B(s)$.

If (ii) holds, the result holds applying the previous case to $A(s)^T$ and $B(s)^T$.

$\Box$

Matrix pencil completion theorems {#seccompletion}
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According to Lemma \[lemmaeq\], the Problem \[problem\] can be approached as a matrix pencil completion problem. We introduce in this section some results in that area which will be used later. To state them we need some notation and definitions.

Given two integers $n$ and $m$, whenever $n>m$ we take $\sum_{i=n}^{m}=0$. In the same way, if a condition is stated for $n\leq i\leq m$ with $n>m$, we understand that the condition disappears.

Given a sequence of integers $a_1, \dots, a_m$ such that $a_1\geq \dots \geq a_m$ we will write ${\mathbf a}=(a_1, \dots,  a_m)$ and we will take $a_i=\infty$ for $i<1$ and $a_i=-\infty$ for $i>m$. If $a_m\geq 0$, the sequence ${\mathbf a}=(a_1, \dots,  a_m)$ is called a [*partition*]{}.

Given three sequences ${\mathbf d}$, ${\mathbf a}$ and ${\mathbf g}$, we introduce next the concept of generalized majorization.

Let ${\mathbf d}= (d_1, \dots, d_m)$, ${\mathbf a}=(a_1, \dots, a_s)$ and ${\mathbf g}=(g_1, \dots, g_{m+s})$ be sequences of integers such that $d_1 \geq \dots \geq d_m$, $a_1 \geq \dots \geq a_s$, 