Matrix pencil
= Matrix pencil
{title2=$A+\lambda B$}
A matrix pencil is an affine one-parameter family of <matrices>. For square matrices of size $m$, its determinant is a <polynomial> of degree at most $m$. If $B$ is invertible, its leading coefficient is $\det B\ne0$, so at most $m$ distinct parameters give a singular matrix. This translates linear dependence in a family of bases into polynomial root counting.