{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A$ be a [[positive definite|0KX66CZMBB]] $n \\times n$ complex Hermitian matrix.\nThen each [[eigenvalue|0BQAA2BK3Q]] of $A$ is a positive real number.\n\n## Remarks\n\nIf $Ax = \\lambda x$ with $x \\ne 0$, then $0 \u003c x^* A x = \\lambda\\, x^* x$, and $x^* x \u003e 0$.\nThe converse holds: a Hermitian matrix with only positive eigenvalues is positive definite, by the spectral theorem.\nSo $\\det A \u003e 0$ and $\\tr A \u003e 0$ for a positive definite $A$.\n\n## Explanations\n\n### L1 human\n\nA positive definite matrix stretches each of its eigenvector directions by a positive factor.\nIt never flips or crushes a direction.\nSo its eigenvalues, its determinant and its trace are all positive.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15B57",
      "15A18"
    ],
    "name": "Eigenvalues of a positive definite matrix are positive",
    "role": "theorem",
    "sources": [
      {
        "locator": "Eigenvalues",
        "url": "https://en.wikipedia.org/wiki/Definite_matrix"
      }
    ],
    "tag": "0HYC2RHD3M",
    "uses": [
      "0QQCG3JWJ4",
      "0ZND7NQ970"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HYC2RHD3M/",
  "links": [
    {
      "field": "uses",
      "target": "0QQCG3JWJ4"
    },
    {
      "field": "uses",
      "target": "0ZND7NQ970"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HYC2RHD3M.md",
  "msc": [
    "15B57",
    "15A18"
  ],
  "name": "Eigenvalues of a positive definite matrix are positive",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HY/0HYC2RHD3M.md",
  "status": null,
  "tag": "0HYC2RHD3M",
  "uri": "m2://0HYC2RHD3M",
  "url": "https://math2.tamnd.com/wiki/eigenvalues-of-a-positive-definite-matrix-are-positive/",
  "writing": {
    "tag": "0HYC2RHD3M",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}