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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be a [[field|04C5ZBG1WT]], let $m, n \\ge 1$, and let $A$ and $B$ be $m \\times n$ [[matrices|0VD9ZSWWJG]] over $\\mathbb{K}$.\nThe *matrix pencil* of $A$ and $B$ is the matrix $A - \\lambda B$ with entries in the polynomial ring $\\mathbb{K}[\\lambda]$.\nWhen $m = n$, the pencil is *regular* if the polynomial $\\det(A - \\lambda B)$ is not the zero polynomial, and *singular* otherwise.\nFor a regular pencil, an *eigenvalue* is a root $\\lambda_0$ of $\\det(A - \\lambda B)$, and an *eigenvector* for $\\lambda_0$ is a vector $v \\ne 0$ with $Av = \\lambda_0 Bv$.\n\n## Remarks\n\nSome authors write $\\lambda B - A$ or $A + \\lambda B$; this changes only signs.\nWhen $B$ is singular, a regular pencil is often said to have an eigenvalue $\\infty$, since $\\mu A - B$ is singular at $\\mu = 0$.\nExample: $A = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 6 \\end{pmatrix}$ and $B = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 2 \\end{pmatrix}$ give $\\det(A - \\lambda B) = (2 - \\lambda)(6 - 2\\lambda)$, so the pencil is regular with eigenvalues $2$ and $3$.\nNon-example: $A = B = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ give $\\det(A - \\lambda B) = (1 - \\lambda) \\cdot 0 = 0$, so the pencil is singular.\nIf $B$ is invertible, the eigenvalues of the pencil are the ordinary [[eigenvalues|0BQAA2BK3Q]] of $B^{-1}A$.\n",
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