Source: https://guthlabs.ai/docs/instruments/eigenvalue-eigenvector-visualizer-2d/mechanism/

instrument mechanism / verified-clean

# Eigenvalue/Eigenvector Visualizer (2D) mechanism

The inspectable method record for Eigenvalue/Eigenvector Visualizer (2D), kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/eigenvalue-eigenvector-visualizer-2d/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/eigenvalue-eigenvector-visualizer-2d/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/eigenvalue-eigenvector-visualizer-2d/qualification/>)

## 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`