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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A = (a_{ij})$ be a real $n \\times n$ [[matrix|0JXZKX526R]].\nThe matrix $A$ is a *Z-matrix* if $a_{ij} \\le 0$ for all $i \\ne j$.\nThe matrix $A$ is an *M-matrix* if it can be written as $A = sI - B$, where:\n\n1. $I$ is the $n \\times n$ [[identity matrix|06DW9BAWB1]] and $s$ is a real number.\n2. $B$ is a [[nonnegative matrix|JPQCCFCY8X]].\n3. $s \\ge \\rho(B)$, where $\\rho(B)$ is the [[spectral radius|9V7YQA85PC]] of $B$.\n\nAn M-matrix is *nonsingular* if moreover $s \u003e \\rho(B)$.\n\n## Remarks\n\nEvery M-matrix is a Z-matrix.\nFor a Z-matrix the choice of $s$ does not matter: whenever $s \\ge \\max_i a_{ii}$, the number $s - \\rho(sI - A)$ is the same.\nExample: $A = \\begin{pmatrix} 2 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix} = 2I - B$ with $B = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\nThe eigenvalues of $B$ are $\\pm 1$, so $\\rho(B) = 1 \u003c 2$, and $A$ is a nonsingular M-matrix.\nNon-example: $A = \\begin{pmatrix} 1 \u0026 -2 \\\\ -2 \u0026 1 \\end{pmatrix} = I - B$ with $B = \\begin{pmatrix} 0 \u0026 2 \\\\ 2 \u0026 0 \\end{pmatrix}$.\nHere $\\rho(B) = 2 \u003e 1$, so clause 3 fails; indeed $A$ has the negative eigenvalue $-1$.\nFor a Z-matrix $A$, each of the following conditions is equivalent to $A$ being a nonsingular M-matrix.\n\n1. $A$ is invertible and $A^{-1}$ is nonnegative.\n2. Every [[eigenvalue|0BQAA2BK3Q]] of $A$ has positive real part.\n3. All leading principal minors of $A$ are positive.\n",
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