{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "312GG23DN4"
    },
    {
      "field": "uses",
      "target": "ZFSZCMKJD1"
    }
  ],
  "body": "\n## Summary\n\nThe Sylvester equation is the linear matrix equation $AX + XB = C$, in which the matrices $A$, $B$, $C$ are given and the matrix $X$ is unknown.\nIt has a unique solution exactly when $A$ and $-B$ have no common [[eigenvalue|0BQAA2BK3Q]].\nIt appears in block diagonalisation, in control theory, and as the general form of the [[Lyapunov equation|92GVPZH9FT]].\n\n## Explanations\n\n### L1 human\n\nThe equation asks for a matrix $X$ such that multiplying it by $A$ on the left and by $B$ on the right, and adding, gives $C$.\nAlthough $X$ is a matrix, the equation is linear in its entries, so it is a large system of ordinary linear equations.\nFor $1 \\times 1$ matrices it reads $ax + xb = c$, with the solution $x = c / (a + b)$ when $a + b \\ne 0$.\nThe condition $a + b \\ne 0$ becomes the condition on eigenvalues for larger matrices.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "15A24",
      "65F45"
    ],
    "name": "Sylvester equation",
    "same_as": {
      "wikidata": "Q3730848",
      "wikipedia": "Sylvester equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sylvester_equation"
      }
    ],
    "tag": "M9SW4NVP2H"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/M9SW4NVP2H/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/M9SW4NVP2H.md",
  "msc": [
    "15A24",
    "65F45"
  ],
  "name": "Sylvester equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M9S/M9SW4NVP2H.md",
  "status": null,
  "tag": "M9SW4NVP2H",
  "uri": "m2://M9SW4NVP2H",
  "url": "https://math2.tamnd.com/wiki/sylvester-equation/",
  "writing": {
    "tag": "M9SW4NVP2H",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}