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  "body": "\n## Summary\n\nAn M-matrix is a real square matrix of the form $sI - B$, where $B$ is a [[nonnegative matrix|JPQCCFCY8X]].\nThe number $s$ must be at least the [[spectral radius|9V7YQA85PC]] of $B$.\nSince $B$ has no negative entry, the entries of an M-matrix off the diagonal are nonpositive.\nSuch matrices appear when a linear system is discretised, in economic input-output models and in Markov chains.\nA nonsingular M-matrix has an inverse with all entries nonnegative, so it turns nonnegative data into nonnegative solutions.\n\n## Explanations\n\n### L1 human\n\nThink of a matrix with positive numbers on the diagonal and negative or zero numbers elsewhere.\nIt is an M-matrix when the diagonal is strong enough to dominate the rest in a precise sense.\nThe matrix $\\begin{pmatrix} 2 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix}$, which comes from the second difference of a function, is an example.\nIts inverse $\\frac{1}{3}\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ has only nonnegative entries.\n",
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