Matrix pencil

In linear algebra, a matrix pencil is a matrix-valued function defined on a field ${\displaystyle K}$, usually the real or complex numbers.

Let ${\displaystyle K}$ be a field (typically, ${\displaystyle K\in \{\mathbb {R} ,\mathbb {C} \}}$; the definition can be generalized to rngs, i.e. non-unital rings), and let ${\displaystyle n>0}$ be a positive integer. Then any matrix-valued function

${\displaystyle P\colon K\to \mathrm {Mat} (K,n\times n)}$

(where ${\displaystyle \mathrm {Mat} (K,n\times n)}$ denotes the ${\displaystyle K}$-algebra of ${\displaystyle n\times n}$ matrices over ${\displaystyle K}$) is called a matrix pencil.

An important special case arises when ${\displaystyle P}$ is polynomial: let ${\displaystyle \ell \geq 0}$ be a non-negative integer, and let ${\displaystyle A_{0},A_{1},\dots ,A_{\ell }}$ be ${\displaystyle n\times n}$ matrices (i. e. ${\displaystyle A_{i}\in \mathrm {Mat} (K,n\times n)}$ for all ${\displaystyle i=0,\dots ,\ell }$). Then the polynomial matrix pencil (often simply a matrix pencil) defined by ${\displaystyle A_{0},\dots ,A_{\ell }}$ is the matrix-valued function ${\displaystyle L\colon K\to \mathrm {Mat} (K,n\times n)}$ defined by

${\displaystyle L(\lambda )=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.}$

The degree of this matrix pencil is defined as the largest integer ${\displaystyle 0\leq k\leq \ell }$ such that ${\displaystyle A_{k}\neq 0}$, the ${\displaystyle n\times n}$ zero matrix over ${\displaystyle K}$.

A particular case is a linear matrix pencil ${\displaystyle L(\lambda )=A-\lambda B}$ (where ${\displaystyle B\neq 0}$).¹ We denote it briefly with the notation ${\displaystyle (A,B)}$, and note that using the more general notation, ${\displaystyle A_{0}=A}$ and ${\displaystyle A_{1}=-B}$ (not ${\displaystyle B}$).

For a matrix pencil ${\displaystyle P}$, any ${\displaystyle k\in K}$ such that ${\displaystyle \det P(k)=0_{K}}$ is called a generalized eigenvalue (often simply eigenvalue) of ${\displaystyle P}$, and the set of generalized eigenvalues of ${\displaystyle P}$ is called its spectrum and is denoted by

${\displaystyle \sigma (P)=\{k\in K:\det P(k)=0_{K}\}.}$

For a polynomial matrix pencil, we write ${\displaystyle \sigma (A_{0},\dots ,A_{\ell })}$; for the linear pencil ${\displaystyle (A,B)}$, we write as ${\displaystyle \sigma (A,B)}$ (not ${\displaystyle \sigma (A,-B)}$).

The generalized eigenvalues of the linear matrix pencil ${\displaystyle (A,I)}$ are precisely the matrix eigenvalues of ${\displaystyle A}$. The general linear pencil ${\displaystyle (A,B)}$ is said to have one or more eigenvalues at infinity if ${\displaystyle B}$ has one or more 0 eigenvalues.

A pencil is called regular if there is at least one ${\displaystyle k\in K}$ such that ${\displaystyle \det P(k)\neq 0_{K}}$, i. e. if ${\displaystyle \lambda (P)\neq K}$; otherwise it is called singular.

Matrix pencils play an important role in numerical linear algebra. The problem of finding the generalized eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, an implicit version of the QR algorithm for solving the eigenvalue problem ${\displaystyle Ax=\lambda Bx}$ without inverting the matrix ${\displaystyle B}$ (which is impossible when ${\displaystyle B}$ is singular, or numerically unstable when it is ill-conditioned).

If ${\displaystyle AB=BA}$, then the pencil generated by ${\displaystyle A}$ and ${\displaystyle B}$:²

1. consists only of matrices similar to a diagonal matrix, or
2. has no matrices in it similar to a diagonal matrix, or
3. has exactly one matrix in it similar to a diagonal matrix.