{
  "title": "Eigenvalue visualizer",
  "human_url": "https://happytails.ai/en/calculators/eigenvalue-visualizer/",
  "agent_url": "https://happytails.ai/agents/en/calculators/eigenvalue-visualizer/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/eigenvalue-visualizer.md",
  "content": "Eigenvalue visualizer\nFind the exact characteristic polynomial of a square matrix of fractions. Rational and real quadratic eigenvalues include exact eigenvectors; higher-degree real roots are isolated exactly, and higher-degree complex roots are numerical approximations.\nHow to use Eigenvalue visualizer\nPaste a square matrix, one row per line, such as 4 1 and 2 3.\nRun the visualizer to compute the characteristic polynomial and factor it.\nRead the eigenvalue kinds, multiplicities, supported eigenvectors and diagonalizability status. A null status means the method did not determine it.\nDownload the JSON report, the CSV of eigenvalues or the plot.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nUseful next steps\nMatrix workspace\nLinear system solver\nGraphing calculator\nSquare matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.\nExact method\nThe characteristic polynomial det(tI - A) is computed with the Faddeev-LeVerrier recurrence in exact fractions and checked against the determinant. It is factored over the rationals. Rational and real quadratic roots give exact eigenvectors from the null space of A - tI, verified by multiplication. Eigenvectors for other root kinds are not computed. Method: Strang, Introduction to Linear Algebra, 5th edition, chapter 6 (2016); Faddeev and Faddeeva, Computational Methods of Linear Algebra (1963).\nHigher-degree factors\nReal roots of irreducible cubic or higher factors are isolated exactly with Sturm sequences and shown with a decimal value. Complex roots of those factors are numerical approximations with polynomial residuals. A small residual checks a candidate root; it does not prove that every complex root was found or that a complete eigenbasis was computed.\nPlot\nFor a 2 by 2 matrix with real eigenvalues the plot shows the unit circle, its image and the invariant directions. Otherwise it shows the eigenvalues in the complex plane.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Matrix workspace",
      "url": "https://happytails.ai/en/calculators/matrix-workspace/"
    },
    {
      "name": "Linear system solver",
      "url": "https://happytails.ai/en/calculators/linear-system-solver/"
    },
    {
      "name": "Graphing calculator",
      "url": "https://happytails.ai/en/calculators/graphing-calculator/"
    }
  ],
  "tool": {
    "id": "eigenvalue-visualizer",
    "category": "calculators",
    "built": true,
    "requirements": "Square matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.",
    "implementation": {
      "fields": [],
      "presentation": "metrics",
      "inputCodeDirection": "ltr",
      "registrationFamily": "exact-math-discrete",
      "inputMaxLength": 20000,
      "inputErrorCode": "EXACT_MATH_DISCRETE_MATRIX_TOO_LONG",
      "note": "Finds the characteristic polynomial and supported exact eigenpairs on your device for square matrices up to 8 by 8. Higher-degree complex roots are numerical approximations.",
      "experience": {
        "automatic": false,
        "single": false,
        "inputMaxLength": 20000,
        "inputMode": "text",
        "label": "Matrix",
        "placeholder": "4 1\n2 3",
        "help": "One row per line or separated by semicolons; entries such as 3, -0.5 or 2/3. Up to 20,000 characters. Review the settings before running.",
        "helpParts": [
          "One row per line or separated by semicolons; entries such as 3, -0.5 or 2/3.",
          "Up to 20,000 characters.",
          "Review the settings before running."
        ],
        "action": null
      },
      "mode": "explicit",
      "interactive": false
    },
    "execution": "Browser interface",
    "api": null,
    "browser_agent": {
      "supported": true,
      "name": "run_current_tool",
      "discovery": "WebMCP on the human page in a supporting browser",
      "verification": "See audit/API-AUDIT.md; availability is not verification"
    }
  },
  "markdown": "# Eigenvalue visualizer\n\n- Human page: https://happytails.ai/en/calculators/eigenvalue-visualizer/\n- Markdown: https://happytails.ai/en/calculators/eigenvalue-visualizer.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/eigenvalue-visualizer/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `eigenvalue-visualizer` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nFinds the characteristic polynomial and supported exact eigenpairs on your device for square matrices up to 8 by 8. Higher-degree complex roots are numerical approximations.\n\n### Inputs and settings\n\nText input: Matrix.\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\n4 1\n2 3\n```\n\nOne row per line or separated by semicolons; entries such as 3, -0.5 or 2/3. Up to 20,000 characters. Review the settings before running.\n\nNo additional settings are declared for this tool.\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nSquare matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"eigenvalue-visualizer\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"requirements\": \"Square matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.\",\n  \"implementation\": {\n    \"fields\": [],\n    \"presentation\": \"metrics\",\n    \"inputCodeDirection\": \"ltr\",\n    \"registrationFamily\": \"exact-math-discrete\",\n    \"inputMaxLength\": 20000,\n    \"inputErrorCode\": \"EXACT_MATH_DISCRETE_MATRIX_TOO_LONG\",\n    \"note\": \"Finds the characteristic polynomial and supported exact eigenpairs on your device for square matrices up to 8 by 8. Higher-degree complex roots are numerical approximations.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": false,\n      \"inputMaxLength\": 20000,\n      \"inputMode\": \"text\",\n      \"label\": \"Matrix\",\n      \"placeholder\": \"4 1\\n2 3\",\n      \"help\": \"One row per line or separated by semicolons; entries such as 3, -0.5 or 2/3. Up to 20,000 characters. Review the settings before running.\",\n      \"helpParts\": [\n        \"One row per line or separated by semicolons; entries such as 3, -0.5 or 2/3.\",\n        \"Up to 20,000 characters.\",\n        \"Review the settings before running.\"\n      ],\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nFind the exact characteristic polynomial of a square matrix of fractions. Rational and real quadratic eigenvalues include exact eigenvectors; higher-degree real roots are isolated exactly, and higher-degree complex roots are numerical approximations.\n\n### How to use Eigenvalue visualizer\n\n1. Paste a square matrix, one row per line, such as 4 1 and 2 3.\n2. Run the visualizer to compute the characteristic polynomial and factor it.\n3. Read the eigenvalue kinds, multiplicities, supported eigenvectors and diagonalizability status. A null status means the method did not determine it.\n4. Download the JSON report, the CSV of eigenvalues or the plot.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n## Useful next steps\n\n- [Matrix workspace](https://happytails.ai/en/calculators/matrix-workspace/) ([Markdown](https://happytails.ai/en/calculators/matrix-workspace.md))\n\n- [Linear system solver](https://happytails.ai/en/calculators/linear-system-solver/) ([Markdown](https://happytails.ai/en/calculators/linear-system-solver.md))\n\n- [Graphing calculator](https://happytails.ai/en/calculators/graphing-calculator/) ([Markdown](https://happytails.ai/en/calculators/graphing-calculator.md))\n\nSquare matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.\n\n## Exact method\n\nThe characteristic polynomial det(tI - A) is computed with the Faddeev-LeVerrier recurrence in exact fractions and checked against the determinant. It is factored over the rationals. Rational and real quadratic roots give exact eigenvectors from the null space of A - tI, verified by multiplication. Eigenvectors for other root kinds are not computed. Method: Strang, Introduction to Linear Algebra, 5th edition, chapter 6 (2016); Faddeev and Faddeeva, Computational Methods of Linear Algebra (1963).\n\n### Higher-degree factors\n\nReal roots of irreducible cubic or higher factors are isolated exactly with Sturm sequences and shown with a decimal value. Complex roots of those factors are numerical approximations with polynomial residuals. A small residual checks a candidate root; it does not prove that every complex root was found or that a complete eigenbasis was computed.\n\n### Plot\n\nFor a 2 by 2 matrix with real eigenvalues the plot shows the unit circle, its image and the invariant directions. Otherwise it shows the eigenvalues in the complex plane.\n"
}