Calculator Inputs
Example Data Table
Sample matrix used by default: [[4, 1, 0], [2, 3, 1], [0, 1, 2]].
| Example | Approximate Eigenvalue | Approximate Normalized Left Eigenvector | Interpretation |
|---|---|---|---|
| Largest mode | 5.114908 | [0.862973, 0.481068, 0.154440] | Dominant left mode with the strongest weighting on the first state. |
| Middle mode | 2.745898 | [0.690080, -0.432715, -0.580126] | Balanced mode showing opposite directional contributions across components. |
| Smallest mode | 1.139194 | [0.414966, -0.593569, 0.689550] | Lower-growth mode emphasizing the third state after transpose analysis. |
Formula Used
Left eigenvector definition: wTA = λwT
Equivalent column form: ATw = λw
Residual vector: r = ATw − λw
Residual norm: ‖r‖2 = √(Σri2)
The page first estimates eigenvalues numerically with a shifted QR process. Each estimated eigenvalue is then refined against the transposed matrix using inverse iteration. The reported vector is normalized according to your selected mode, and the residual norm measures how closely the computed vector satisfies the left eigenvector equation.
How to Use This Calculator
- Select a matrix size from 2 × 2 up to 5 × 5.
- Enter every matrix coefficient, or load the default sample.
- Choose display precision, normalization style, iteration limit, and tolerance.
- Optionally enter a target eigenvalue to highlight the closest computed left eigenpair.
- Press Calculate Left Eigenvectors to show the results above the form.
- Review eigenvalues, normalized left eigenvectors, residual vectors, and residual norms.
- Inspect the Plotly chart to compare vector components and residual behavior.
- Use the CSV or PDF buttons to export the current computed table.
Frequently Asked Questions
1) What is a left eigenvector?
A left eigenvector is a row vector wT that satisfies wTA = λwT. It describes how a matrix acts from the left side and is equivalent to solving ATw = λw.
2) How is it different from a right eigenvector?
A right eigenvector satisfies Av = λv. A left eigenvector satisfies ATw = λw. For symmetric matrices they are closely related, but for general matrices they can differ significantly.
3) Why does the calculator use the transpose?
Because solving for left eigenvectors directly as rows is equivalent to solving a standard right-eigenvector problem on the transposed matrix. That turns the left problem into a familiar column-vector computation.
4) What does the residual norm mean?
The residual norm measures numerical accuracy. Smaller values mean the reported vector more closely satisfies ATw = λw. Values near zero indicate a stronger computed eigenpair.
5) Which normalization should I choose?
Unit length is best for comparison. First nonzero component = 1 is useful for algebra notes. Largest absolute component = 1 is convenient when comparing relative weights among entries.
6) Can this handle complex eigenvalues?
This page targets real-valued numeric outputs. If your matrix has complex eigenvalues, the reported values can be approximate or incomplete. For exact complex analysis, use a symbolic or complex-number library.
7) Why can repeated eigenvalues be tricky?
Repeated eigenvalues may correspond to multiple valid eigenvectors or poorly separated directions. Numerical methods can return one stable basis direction rather than every independent vector unless extra decomposition steps are used.
8) When are left eigenvectors useful?
They appear in control theory, Markov modeling, sensitivity analysis, modal analysis, and non-symmetric systems. They help measure weighting, observability, and directional influence in transformed coordinates.