happytails.ai

Eigenvalue visualizer

Open Eigenvalue visualizer on Happytails.ai. This view has the same page content and links, with explicit tool capabilities. Read the processing notes before using a tool.

Tool capabilities

{
  "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"
  }
}

Page guide

Find 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.

How to use Eigenvalue visualizer
  1. Paste a square matrix, one row per line, such as 4 1 and 2 3.
  2. Run the visualizer to compute the characteristic polynomial and factor it.
  3. Read the eigenvalue kinds, multiplicities, supported eigenvectors and diagonalizability status. A null status means the method did not determine it.
  4. Download the JSON report, the CSV of eigenvalues or the plot.

Useful next steps

Matrix workspaceLinear system solverGraphing calculator

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.

Exact method

The 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).

Higher-degree factors

Real 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.

Plot

For 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.