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  "body": "\n## Summary\n\nA matrix pencil is a family of matrices $A - \\lambda B$ that depends linearly on a parameter $\\lambda$.\nIts eigenvalues are the values of $\\lambda$ for which $A - \\lambda B$ is singular, which leads to the generalised eigenvalue problem $Av = \\lambda Bv$.\nPencils arise in vibration problems, in differential-algebraic equations and in control theory, and they are classified by the Kronecker canonical form.\nMore general pencils allow higher powers of $\\lambda$; the linear ones are the most common.\n\n## Explanations\n\n### L1 human\n\nTake two matrices $A$ and $B$ of the same size and look at all the combinations $A - \\lambda B$ as the number $\\lambda$ varies.\nThe ordinary eigenvalue problem is the case $B = I$.\nIn mechanics, $A$ may describe stiffness and $B$ mass, and the eigenvalues of the pencil give the squares of the natural frequencies.\nFor $A = \\operatorname{diag}(1, 2)$ and $B = I$, the matrix $A - \\lambda B$ is singular exactly for $\\lambda = 1$ and $\\lambda = 2$.\n",
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