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Suppose $M$ is a square matrix with full rank. If $v$ and $w$ are column vectors, then the expression $$M^Tvw^TM =: A$$ is a matrix.

Under what assumptions on $v$ and $w$ can we say that $A$ has positive entries? I don't know if we can say anything about entries but one can hope.

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Alternatively we could write $M^TBM=A$ where $B$ is a matrix of rank $1$. But where do we put quantifiers? Is the question "does there exist $B$ such that $A$ is entrywise-positive for all $M$" or "given $M$, does there exist $B$ such that $A$ is entrywise-positive"? – user53153 Dec 25 '12 at 19:22
@PavelM Your second question is the intended one, $M$ is given. – Lemon Dec 26 '12 at 11:56
up vote 1 down vote accepted

Since $M$ is invertible, there exists a vector $v$ such that $M^Tv=(1,1,\dots,1)^T$. Then $v^TM=(1,1,\dots,1)$, and $M^Tvv^TM=(1,1,\dots,1)^T(1,1,\dots,1)$, which is a matrix with all entries equal to $1$.

More generally, if $M^Tv$ and $w^TM$ have positive entries (and since $M$ is invertible, you can tell precisely for which vectors $v$ and $w$ this holds), the product matrix has positive entries.

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