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.