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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $A$ be a real $n \\times n$ [[nonnegative matrix|JPQCCFCY8X]], and let $r = \\rho(A)$ be its [[spectral radius|9V7YQA85PC]].\n\n1. The number $r$ is an [[eigenvalue|0BQAA2BK3Q]] of $A$, and it has an [[eigenvector|0N6960W0BV]] $v \\ne 0$ with all entries nonnegative.\n2. If $n \\ge 2$ and $A$ is [[irreducible|7Q60BB01YG]], then $r \u003e 0$ and $r$ is a simple root of the [[characteristic polynomial|0VSSPAFM2C]] of $A$.\n3. If $n \\ge 2$ and $A$ is irreducible, then $r$ has an eigenvector $v$ with all entries positive.\n4. If $n \\ge 2$ and $A$ is irreducible, then every nonnegative eigenvector of $A$ is a positive multiple of $v$.\n5. If $n \\ge 2$, $A$ is irreducible, and $h$ is the number of eigenvalues of $A$ of absolute value $r$, then these eigenvalues are $r e^{2 \\pi i k / h}$ for $k = 0, 1, \\ldots, h - 1$.\n6. If all entries of $A$ are positive, then the conclusions of 2, 3 and 4 hold, and $\\abs{\\lambda} \u003c r$ for every eigenvalue $\\lambda \\ne r$ of $A$.\n\n## Remarks\n\nOne proof of part 6 looks at the map $x \\mapsto Ax / \\sum_i (Ax)_i$ on the simplex of nonnegative vectors with entry sum $1$; Brouwer's fixed point theorem gives a positive eigenvector.\nFor an irreducible matrix, another proof uses the Collatz–Wielandt formula $r = \\max_{x \\ge 0,\\, x \\ne 0} \\min_{x_i \u003e 0} (Ax)_i / x_i$.\nPart 1 follows from part 6 by continuity, applied to $A + \\varepsilon J$ where $J$ has all entries $1$ and $\\varepsilon \\to 0$.\nExample: $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is irreducible with $r = 1$, eigenvector $(1, 1)$, and eigenvalues $1$ and $-1$, so $h = 2$; the strict inequality of part 6 fails because $A$ has zero entries.\nWithout irreducibility, simplicity can fail: the identity matrix $I_2$ has $r = 1$ as a double eigenvalue.\nFor the transition matrix of a finite irreducible [[Markov chain|0G48PYM96Q]], the theorem gives a unique stationary distribution.\n\n## Explanations\n\n### L1 human\n\nA matrix with nonnegative entries has a special eigenvalue: its spectral radius, the largest absolute value of an eigenvalue, is itself an eigenvalue, a real number $r \\ge 0$.\nIts eigenvector can be chosen with nonnegative entries, so it makes sense as a list of sizes, shares or probabilities.\nWhen all entries are positive, this eigenvalue strictly beats all the others, and repeated multiplication by $A$ pushes every positive vector towards the direction of that eigenvector.\nThis is why rankings such as PageRank and long-run behaviour of population models are computed from it.\n\n### L2 human\n\nThe theorem has three layers: nonnegative matrices give a nonnegative eigenvector for $\\rho(A)$; irreducible ones give simplicity and positivity; positive ones give a spectral gap.\nThe number $h$ in part 5 is the period of the directed graph of $A$, and $A$ is called primitive when $h = 1$.\nFor a primitive matrix, $A^k / r^k$ converges to the rank one matrix $v w^T / (w^T v)$, where $w$ is the positive left eigenvector.\nThe theorem underlies the convergence of finite Markov chains, the [[M-matrix|DNKXHMGQY1]] theory and power iteration for the dominant eigenvalue.\n",
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