Matrix pencil

Let $K$ be a field (typically, $K\; \backslash in\; \backslash \{\; \backslash mathbb\; R,\; \backslash mathbb\; C\; \backslash \}$; the definition can be generalized to rngs), let $\backslash ell\; \backslash ge\; 0$ be a non-negative integer, let $n\; >\; 0$ be a positive integer, and let $A\_0,\; A\_1,\; \backslash dots,\; A\_\backslash ell$ be $n\backslash times\; n$ matrices (i. e. $A\_i\; \backslash in\; \backslash mathrm\{Mat\}(K,\; n\; \backslash times\; n)$ for all $i\; =\; 0,\; \backslash dots,\; \backslash ell$). Then the matrix pencil defined by $A\_0,\; \backslash dots,\; A\_\backslash ell$ is the matrix-valued function $L\; \backslash colon\; K\; \backslash to\; \backslash mathrm\{Mat\}(K,\; n\; \backslash times\; n)$ defined by

:$L(\backslash lambda)\; =\; \backslash sum\_\{i=0\}^\backslash ell\; \backslash lambda^i\; A\_i.$

The degree of the matrix pencil is defined as the largest integer $0\; \backslash le\; k\; \backslash le\; \backslash ell$ such that $A\_k\; \backslash ne\; 0$, the $n\; \backslash times\; n$ zero matrix over $K$.

A particular case is a linear matrix pencil $L(\backslash lambda)\; =\; A\; -\; \backslash lambda\; B$ (where $B\; \backslash ne\; 0$).{{harvtxt|Golub|Van Loan|1996|p=375}} We denote it briefly with the notation $(A,\; B)$, and note that using the more general notation, $A\_0\; =\; A$ and $A\_1\; =\; -B$ (not $B$).

A pencil is called regular if there is at least one value of $\backslash lambda$ such that $\backslash det(L(\backslash lambda))\; \backslash neq\; 0$; otherwise it is called singular. We call eigenvalues of a matrix pencil all (complex) numbers $\backslash lambda$ for which $\backslash det(L(\backslash lambda))\; =\; 0$; in particular, the eigenvalues of the matrix pencil $(A,\; I)$ are the matrix eigenvalues of $A$. For linear pencils in particular, the eigenvalues of the pencil are also called generalized eigenvalues.

The set of the eigenvalues of a pencil is called the spectrum of the pencil, and is written $\backslash sigma(A\_0,\; \backslash dots,\; A\_\backslash ell)$. For the linear pencil $(A,\; B)$, it is written as $\backslash sigma(A,\; B)$ (not $\backslash sigma(A,\; -B)$).

The linear pencil $(A,\; B)$ is said to have one or more eigenvalues at infinity if $B$ has one or more 0 eigenvalues.

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

If $AB\; =\; BA$, then the pencil generated by $A$ and $B$:{{harvtxt|Marcus & Minc|1969|p=79}}

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.

• Generalized eigenvalue problem
• Generalized pencil-of-function method
• Nonlinear eigenproblem
• Quadratic eigenvalue problem
• Generalized Rayleigh quotient

{{Reflist}}

• {{ citation | first1 = Gene H. | last1 = Golub | first2 = Charles F. | last2 = Van Loan | year = 1996 | isbn = 0-8018-5414-8 | title = Matrix Computations | edition = 3rd | publisher = Johns Hopkins University Press | location = Baltimore }}
• {{Citation

| last = Marcus & Minc

| date = 1969

| title = A survey of matrix theory and matrix inequalities

| publisher = Courier Dover Publications

}}

• Peter Lancaster & Qian Ye (1991) "Variational and numerical methods for symmetric matrix pencils", Bulletin of the Australian Mathematical Society 43: 1 to 17

Category:Linear algebra

{{Linear-algebra-stub}}