{
    "name": "Matrix Eigenvalues Calculator",
    "url": "https://calculators.geekmidst.com/calculators/matrix-eigenvalues-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/matrix-eigenvalues-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find all eigenvalues of a square matrix up to 10 × 10, real and complex, with the exact characteristic polynomial and exact multiplicities.",
    "summary": "Type the matrix with one row per line. The characteristic polynomial det(λI − A) is computed exactly, and its roots are the eigenvalues, complex ones included.",
    "also_called": [
        "Characteristic polynomial calculator",
        "Eigenvalue finder"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "matrix",
            "label": "Matrix A",
            "type": "textarea",
            "default": "4 1 2\n0 3 1\n1 0 2"
        }
    ],
    "outputs": [
        {
            "key": "eigenvalues",
            "label": "Eigenvalues λ",
            "format": "text"
        },
        {
            "key": "char_poly",
            "label": "Characteristic polynomial det(λI − A)",
            "format": "text"
        },
        {
            "key": "trace",
            "label": "Trace (sum of eigenvalues)",
            "format": "text"
        },
        {
            "key": "determinant",
            "label": "Determinant (product of eigenvalues)",
            "format": "text"
        },
        {
            "key": "spectral_radius",
            "label": "Spectral radius max |λ|",
            "format": "number"
        },
        {
            "key": "real_count",
            "label": "Real eigenvalues (with multiplicity)",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "p(λ) = det(λI − A) by the Faddeev–LeVerrier recursion: Mₖ = A·Mₖ₋₁ + cₙ₋ₖ₊₁·I,  cₙ₋ₖ = −tr(A·Mₖ) ÷ k;  eigenvalues are the roots of p(λ)",
        "variables": {
            "A": "the n × n matrix",
            "I": "the identity matrix",
            "cₖ": "coefficients of the characteristic polynomial",
            "λ": "an eigenvalue"
        }
    },
    "worked_example": {
        "inputs": {
            "matrix": "4 1 2\n0 3 1\n1 0 2"
        },
        "steps": [
            "Trace = 4 + 3 + 2 = 9 and det A = 4(3·2 − 1·0) − 1(0·2 − 1·1) + 2(0·0 − 3·1) = 24 + 1 − 6 = 19.",
            "The sum of the 2 × 2 principal minors is (12 − 0) + (8 − 2) + (6 − 0) = 24, so p(λ) = λ³ − 9λ² + 24λ − 19.",
            "Its roots are λ ≈ 1.4679, 2.6527 and 4.8794; they add to 9 and multiply to 19."
        ],
        "result": "The matrix has three real eigenvalues, about 1.468, 2.653 and 4.879."
    },
    "interpretation": "Eigenvalues tell you how the matrix stretches space along its eigenvectors. Complex eigenvalues of a real matrix come in conjugate pairs and indicate rotation. The spectral radius decides whether repeated multiplication by A grows (> 1) or dies out (< 1).",
    "assumptions": [
        "Entries are real numbers, read exactly as decimals or fractions."
    ],
    "limitations": [
        "Up to 10 × 10.",
        "Eigenvectors are not computed.",
        "Eigenvalues are numerical (10 significant figures); the characteristic polynomial is exact."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "What does multiplicity mean here?",
            "answer": "It is the algebraic multiplicity: how many times λ is a root of the characteristic polynomial. It is computed exactly, so a repeated eigenvalue is not reported as two nearly equal numbers."
        },
        {
            "question": "Why are the eigenvalues of a symmetric matrix real?",
            "answer": "The spectral theorem: a real symmetric matrix can be diagonalised by an orthogonal matrix, so all its eigenvalues are real."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Matrix Calculator",
            "url": "https://calculators.geekmidst.com/calculators/matrix-calculator"
        },
        {
            "name": "Polynomial Roots Calculator",
            "url": "https://calculators.geekmidst.com/calculators/polynomial-roots-calculator"
        },
        {
            "name": "Gaussian Elimination Calculator",
            "url": "https://calculators.geekmidst.com/calculators/linear-system-gaussian-elimination-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}