Random Eigenpair Verification Exercise Generator

Description

Construct diagonal two-dimensional eigenpair verification cards with exact matrix products, explicit false candidates and repeated-eigenvalue labels.

A matrix construction produces numeric arrays with a target algebraic property. Floating-point residuals check how closely a computed result satisfies that property, while conditioning determines how sensitive it is to numerical error.

Construct diagonal two-dimensional eigenpair verification cards with exact matrix products, explicit false candidates and repeated-eigenvalue labels.

When to use this tool

Use this generator to create reproducible eigenpair verification exercise cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply minimum, maximum, invalidProbability, count in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read cards. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Draw a diagonal two-by-two matrix and choose a coordinate basis vector. Use the corresponding diagonal value as the candidate eigenvalue, optionally add one to make it invalid, and compare delegated matrix-vector multiplication with eigenvalue times vector.^[1]

Delegated operations used for this result: multiply. Their calculation or parsing conventions apply to the corresponding result fields.^[2]

Input field	Meaning and units	Accepted input
minimum	Minimum	Required. Safe integer.
maximum	Maximum	Required. Safe integer.
invalidProbability	Invalid Probability	Required. Finite number; minimum 0; maximum 1.
count	Count	Required. Safe integer; minimum 1; maximum 16.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
cards	Cards	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16.
cards[].matrix	Matrix	Required. Array of Array of Safe integer.; minimum items 2; maximum items 2.; minimum items 2; maximum items 2.
cards[].vector	Vector	Required. Array of Safe integer.; minimum items 2; maximum items 2.
cards[].eigenvalue	Eigenvalue	Required. Safe integer.
cards[].product	Product	Required. Array of Safe integer.; minimum items 2; maximum items 2.
cards[].expectedProduct	Expected Product	Required. Array of Safe integer.; minimum items 2; maximum items 2.
cards[].valid	Valid	Required. Boolean.
cards[].repeatedEigenvalue	Repeated Eigenvalue	Required. Boolean.
randomness	Replay source and normalized 32-bit seed, or null seed for an advancing context source.	Required. Object; no unknown keys.
randomness.seed	Optional safe-integer replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "minimum": 1,
    "maximum": 5,
    "invalidProbability": 0,
    "count": 1,
    "seed": 42
  },
  "output": {
    "cards": [
      {
        "matrix": [
          [
            4,
            0
          ],
          [
            0,
            3
          ]
        ],
        "vector": [
          0,
          1
        ],
        "eigenvalue": 3,
        "product": [
          0,
          3
        ],
        "expectedProduct": [
          0,
          3
        ],
        "valid": true,
        "repeatedEigenvalue": false
      }
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Validity uses exact equality on bounded integer products. Repeated eigenvalues are explicitly flagged. The generator covers this simple planted family, not arbitrary eigensolvers or general numerical tolerance choices.
A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Eigenpair Verification Exercise Generator — tool.io
Calculation or validation: matrix/multiply — tool.io

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