Matrix pencil

Remarks

Some authors write $\lambda B - A$ or $A + \lambda B$; this changes only signs. When $B$ is singular, a regular pencil is often said to have an eigenvalue $\infty$, since $\mu A - B$ is singular at $\mu = 0$. Example: $A = \begin{pmatrix} 2 & 0 \\ 0 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ give $\det(A - \lambda B) = (2 - \lambda)(6 - 2\lambda)$, so the pencil is regular with eigenvalues $2$ and $3$. Non-example: $A = B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ give $\det(A - \lambda B) = (1 - \lambda) \cdot 0 = 0$, so the pencil is singular. If $B$ is invertible, the eigenvalues of the pencil are the ordinary eigenvalues of $B^{-1}A$.