Rank of a linear combination of matrices

Question

Suppose you have $q$ matrices $M_1,M_2,\ldots,M_q$ of the same dimentions $(p\times m)$, with $p\geq m$. Is there a necessary and sufficient condition on the $M_1,M_2,\ldots,M_q$ matrices such that the linear combination \begin{equation} M = \lambda_1 M_1+\lambda_2 M_2 + \cdots + \lambda_q M_q \end{equation} has full column rank, for some scalars $\lambda_1, \lambda_2,\ldots,\lambda_q$?

Answer 1

A $p \times m$ matrix with $p \ge m$ has full rank $m$ if and only if at least one of its $m \times m$ minors doesn't vanish. The $m \times m$ minors of $M$ are polynomials in $\lambda_1, \dots \lambda_q$ of degree $m$, so the condition to check is that these polynomials are not all equal to zero. Whether that's useful to you depends on what you're trying to do.