{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] with real entries.\nThe matrix $A$ is *totally positive* when every [[minor|0BT9BYWVRY]] of $A$ is positive.\nThat is, for all $1 \\le k \\le \\min(m, n)$, all rows $i_1 \u003c \\cdots \u003c i_k$ and all columns $j_1 \u003c \\cdots \u003c j_k$, the $k \\times k$ submatrix on these rows and columns has positive determinant.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ is totally positive: its $1 \\times 1$ minors $1, 1, 1, 2$ are positive, and its only $2 \\times 2$ minor is $2 - 1 = 1 \u003e 0$.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 2 \u0026 1 \\end{pmatrix}$ is not totally positive: all entries are positive, but the determinant is $1 - 4 = -3$.\nThe matrix is *totally nonnegative* when every minor is nonnegative.\nSome authors use \"totally positive\" for this weaker condition and \"strictly totally positive\" for the condition above.\nThe [[Vandermonde matrix|Q3JWBZ7EFY]] of real numbers $0 \u003c x_1 \u003c \\cdots \u003c x_n$ is totally positive.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "5GN0RYRVBA",
    "kind": "definition",
    "msc": [
      "15B48"
    ],
    "name": "Totally positive matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Totally_positive_matrix"
      }
    ],
    "tag": "BTRNZKN1BP",
    "uses": [
      "0VD9ZSWWJG",
      "0BT9BYWVRY"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/BTRNZKN1BP/",
  "links": [
    {
      "field": "defines",
      "target": "5GN0RYRVBA"
    },
    {
      "field": "uses",
      "target": "0VD9ZSWWJG"
    },
    {
      "field": "uses",
      "target": "0BT9BYWVRY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/BTRNZKN1BP.md",
  "msc": [
    "15B48"
  ],
  "name": "Totally positive matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BTR/BTRNZKN1BP.md",
  "status": null,
  "tag": "BTRNZKN1BP",
  "uri": "m2://BTRNZKN1BP",
  "url": "https://math2.tamnd.com/wiki/totally-positive-matrix-definition/",
  "writing": {
    "tag": "BTRNZKN1BP",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}