Calculator Inputs
Example Data Table
Use this example to test the calculator with a simple symmetric matrix.
| Input | Value | Meaning |
|---|---|---|
| Matrix size | 2 x 2 | Two rows and two columns |
| Matrix A | [[2, 1], [1, 2]] | Symmetric test matrix |
| Approximate λ | 3 | Expected dominant eigenvalue |
| Expected vector | [0.7071, 0.7071] | Normalized pseudo eigenvector |
Formula Used
The calculator checks the near eigenvector relation A v ≈ λ v.
It builds B = A - λI and searches for a normalized vector near the null space of B.
For a 2 x 2 matrix, it tests perpendicular row candidates. For a 3 x 3 matrix, it tests cross products of row pairs. Then it selects the vector with the smallest residual.
The main residual is r = A v - λ v.
The residual norm is ||r||₂.
The Rayleigh quotient is ρ = (vᵀAv) / (vᵀv).
How to Use This Calculator
- Select a 2 x 2 or 3 x 3 square matrix.
- Enter every matrix value in the input boxes.
- Enter an approximate real eigenvalue.
- Set a residual tolerance for pass or review status.
- Press the calculate button.
- Read the normalized vector and residual values.
- Use CSV or PDF export for reports.
Pseudo Eigenvectors in Matrix Analysis
What a Pseudo Eigenvector Means
A true eigenvector satisfies a strict rule. Multiplying the matrix by the vector only changes its scale. That scale is the eigenvalue. In real work, the rule is often approximate. Data can contain noise. A matrix can be rounded. A guessed eigenvalue may not be exact. A pseudo eigenvector helps in those cases. It gives a direction that nearly follows the eigenvector rule.
Why Residual Size Matters
The residual tells how far the vector is from the exact relation. A small residual is better. It means the computed direction behaves like an eigenvector for the selected value of λ. A large residual means the eigenvalue may be wrong. It can also mean the matrix has no close direction for that value. The relative residual gives a scaled view. This helps when matrix values are very large or very small.
Practical Uses
Pseudo eigenvectors are useful in numerical linear algebra. They also appear in data science and simulation. They can support stability checks, ranking systems, vibration studies, and approximation methods. They are helpful when exact symbolic work is slow. They are also useful during quick validation. A normalized vector makes comparison easier. It removes scale from the result.
Reading the Output
Start with the vector. Then compare A times v with λ times v. If both lists are close, the direction is strong. Next, check the residual norm. Compare it with your tolerance. The Rayleigh quotient gives another useful estimate. It shows the eigenvalue suggested by the computed vector itself. When λ and the Rayleigh value are close, the result is usually more reliable. Always check the original matrix meaning before using results in design, finance, safety, or engineering decisions.
Frequently Asked Questions
1. What is a pseudo eigenvector?
A pseudo eigenvector is a vector that nearly satisfies A v = λ v. It is useful when the eigenvalue is approximate, the matrix has noise, or exact eigenvectors are hard to obtain.
2. Does this calculator find exact eigenvalues?
No. It uses the eigenvalue you enter. It then finds a normalized vector that best fits that entered value using residual-based candidate testing.
3. What does residual norm mean?
The residual norm measures the size of A v minus λ v. Smaller values mean the vector behaves more like a true eigenvector for your chosen eigenvalue.
4. Why is the vector normalized?
Eigenvectors can be scaled without changing their direction. Normalization gives a unit-length vector, so results are easier to compare and export.
5. Can I use a 3 x 3 matrix?
Yes. Select 3 x 3 in the matrix size menu. The hidden third row and third column fields will appear automatically.
6. What is the Rayleigh quotient?
The Rayleigh quotient estimates the eigenvalue linked with the calculated vector. It is computed as vᵀAv divided by vᵀv.
7. What tolerance should I use?
Use a small tolerance such as 0.000001 for clean numerical examples. Use a larger value when your matrix comes from noisy measurements.
8. Why does status show review?
Review means the residual norm is greater than your tolerance. Try a better eigenvalue estimate or inspect the matrix for scaling and entry errors.