Eigenvalue/Eigenvector Visualizer (2D) mechanism

Model scope

The core characteristic-polynomial and nullspace formulas are standard for 2x2 matrices. Review is required because the real-versus-complex domain, repeated/defective cases, eigenvector canonicalization, and exact plotted vector-field definition are not specified. The honest label should also state whether complex eigenpairs are reported and that a real directional plot may show no invariant directions for matrices such as a 90-degree rotation.

• det(A-λI)=0 is a correct characteristic-equation method, equivalent to det(λI-A)=0 for a 2x2 matrix.
• (A-λI)v=0 is correct only with the explicit constraint v≠0; the zero vector is not an eigenvector.
• The specification does not define handling of repeated eigenvalues or defective matrices, which may have fewer than two linearly independent eigenvectors.
• The specification does not define whether complex eigenvalues/eigenvectors are supported. A real 2D directional plot cannot display complex invariant directions.
• The plotted quantity is underspecified: 'sample vector field showing stretch' does not state whether arrows represent Av, displacement Av-v, normalized directions, or another field.
• Eigenvector normalization, sign/phase convention, and ordering of eigenvalues/eigenvectors are unspecified, so exact plotted outputs may be non-deterministic.

Units and conventions

• Matrix entries are assumed to be real numbers unless complex arithmetic is explicitly supported.
• All matrix entries must use compatible input and output coordinate units. If the matrix maps a coordinate space to itself, eigenvalues are dimensionless stretch factors; otherwise their units are output-units/input-units.
• The sampled vector-field points below interpret the plotted quantity as the transformed vector Av, with no normalization.
• Eigenvectors are represented by any nonzero scalar multiple; the listed representatives use simple real or complex coordinates.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "matrix": [
      [
        2,
        0
      ],
      [
        0,
        3
      ]
    ],
    "sample_vectors": [
      [
        1,
        0
      ],
      [
        0,
        1
      ],
      [
        1,
        1
      ]
    ]
  },
  "expected_outputs": {
    "eigenvalues": [
      2,
      3
    ],
    "eigenvectors": {
      "lambda_2": [
        1,
        0
      ],
      "lambda_3": [
        0,
        1
      ]
    },
    "sampled_transformed_vectors": [
      {
        "input": [
          1,
          0
        ],
        "output": [
          2,
          0
        ]
      },
      {
        "input": [
          0,
          1
        ],
        "output": [
          0,
          3
        ]
      },
      {
        "input": [
          1,
          1
        ],
        "output": [
          2,
          3
        ]
      }
    ],
    "key_curve_features": [
      "Real invariant directions are the x-axis and y-axis.",
      "The origin is fixed.",
      "Vectors on the x-axis are stretched by factor 2; vectors on the y-axis are stretched by factor 3."
    ]
  },
  "working": "A-λI=[[2-λ,0],[0,3-λ]]. det(A-λI)=(2-λ)(3-λ)=0, so λ=2 or λ=3. For λ=2: [[0,0],[0,1]]v=0 gives y=0, so v=(1,0). For λ=3: [[-1,0],[0,0]]v=0 gives x=0, so v=(0,1). Direct products: A(1,0)=(2*1+0*0,0*1+3*0)=(2,0); A(0,1)=(0,3); A(1,1)=(2*1+0*1,0*1+3*1)=(2,3)."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification does not state how repeated eigenvalues with non-unique eigenspaces, zero-denominator solver cases, complex eigenvalues, or eigenvector sign and normalization conventions are handled. These inputs can cause incorrect directions, undefined arithmetic, or an invalid real plot unless explicitly guarded.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json