{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $E$ have order $d$ and coefficients $c_0, \\dots, c_{d-1} \\in R$. It is a [[linear recurrence with constant coefficients|081THG13JZ]]. Let $u_n = q^n$ for $n \\in \\N$, with $q^0 = 1$.\n2. By the definition of the [[characteristic polynomial|0BWDJGQYST]], $\\chi_E(q) = q^d - \\sum_{i=0}^{d-1} c_i q^i$.\n3. For each $n \\in \\N$, since $R$ is commutative,\n$$u_{n+d} - \\sum_{i=0}^{d-1} c_i\\, u_{n+i} = q^n q^d - \\sum_{i=0}^{d-1} c_i\\, q^n q^i = q^n\\, \\chi_E(q).$$\n4. By the definition of a [[solution|0ZV2YKKZ58]], $(u_n)$ solves $E$ exactly when the left side of step 3 is $0$ for every $n \\in \\N$.\n5. Suppose $(u_n)$ is a solution. Step 3 with $n = 0$ gives $0 = q^0 \\chi_E(q) = \\chi_E(q)$. So $q$ is a root of $\\chi_E$.\n6. Conversely, suppose $\\chi_E(q) = 0$. Then step 3 gives $q^n \\chi_E(q) = 0$ for every $n$. By step 4, $(q^n)_{n \\ge 0}$ is a solution of $E$.\n\n## Idea\n\nPlug $q^n$ into the recurrence. Every term has the common factor $q^n$, and what remains is exactly $\\chi_E(q)$.\nAt $n = 0$ the factor is $1$, so the recurrence holds for all $n$ exactly when $\\chi_E(q) = 0$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "39A06"
    ],
    "name": "Proof that geometric solutions come from roots of the characteristic polynomial",
    "proves": "096SD4ERQH",
    "sources": [
      {
        "locator": "Characteristic polynomial and roots",
        "url": "https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients"
      }
    ],
    "tag": "7ZQPD24050",
    "uses": [
      "081THG13JZ",
      "0ZV2YKKZ58",
      "0BWDJGQYST"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/7ZQPD24050/",
  "links": [
    {
      "field": "proves",
      "target": "096SD4ERQH"
    },
    {
      "field": "uses",
      "target": "081THG13JZ"
    },
    {
      "field": "uses",
      "target": "0ZV2YKKZ58"
    },
    {
      "field": "uses",
      "target": "0BWDJGQYST"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7ZQPD24050.md",
  "msc": [
    "39A06"
  ],
  "name": "Proof that geometric solutions come from roots of the characteristic polynomial",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7ZQ/7ZQPD24050.md",
  "status": null,
  "tag": "7ZQPD24050",
  "uri": "m2://7ZQPD24050",
  "url": "https://math2.tamnd.com/wiki/proof-that-geometric-solutions-come-from-roots-of-the-characteristic-polynomial/",
  "writing": {
    "tag": "7ZQPD24050",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}