1
$\begingroup$

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.

  • 0
    @PavelM Your second question is the intended one, $M$ is given.2012-12-26

1 Answers 1

1

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.