"defective matrix" meaning in All languages combined

{
  "forms": [
    {
      "form": "defective matrices",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "defective matrices"
      },
      "expansion": "defective matrix (plural defective matrices)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Linear algebra",
          "orig": "en:Linear algebra",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41,\nAn n x n matrix A is a defective matrix if it has a defective eigenvalue.\nObviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices."
        },
        {
          "ref": "2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261",
          "text": "Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.",
          "type": "quotation"
        },
        {
          "ref": "2013, Angelo Luongo, “Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures”, in Alexander F. Vakakis, editor, Normal Modes and Localization in Nonlinear Systems, Springer, page 153",
          "text": "Thus, defective matrices exhibit high sensitivity to imperfections.\nIt can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_(id).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable."
      ],
      "id": "en-defective_matrix-en-noun-GpET1bu8",
      "links": [
        [
          "linear algebra",
          "linear algebra"
        ],
        [
          "matrix",
          "matrix"
        ],
        [
          "linearly independent",
          "linearly independent"
        ],
        [
          "eigenvector",
          "eigenvector"
        ],
        [
          "diagonalisable",
          "diagonalisable"
        ]
      ],
      "raw_glosses": [
        "(linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable."
      ],
      "related": [
        {
          "word": "characteristic polynomial"
        },
        {
          "word": "defective eigenvalue"
        }
      ],
      "topics": [
        "linear-algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "defective matrix"
}

{
  "forms": [
    {
      "form": "defective matrices",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "defective matrices"
      },
      "expansion": "defective matrix (plural defective matrices)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "characteristic polynomial"
    },
    {
      "word": "defective eigenvalue"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Pages with 1 entry",
        "en:Linear algebra"
      ],
      "examples": [
        {
          "text": "1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41,\nAn n x n matrix A is a defective matrix if it has a defective eigenvalue.\nObviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices."
        },
        {
          "ref": "2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261",
          "text": "Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.",
          "type": "quotation"
        },
        {
          "ref": "2013, Angelo Luongo, “Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures”, in Alexander F. Vakakis, editor, Normal Modes and Localization in Nonlinear Systems, Springer, page 153",
          "text": "Thus, defective matrices exhibit high sensitivity to imperfections.\nIt can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_(id).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable."
      ],
      "links": [
        [
          "linear algebra",
          "linear algebra"
        ],
        [
          "matrix",
          "matrix"
        ],
        [
          "linearly independent",
          "linearly independent"
        ],
        [
          "eigenvector",
          "eigenvector"
        ],
        [
          "diagonalisable",
          "diagonalisable"
        ]
      ],
      "raw_glosses": [
        "(linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable."
      ],
      "topics": [
        "linear-algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "defective matrix"
}