Calculator Form
Enter a square matrix with rows on new lines. Separate values with spaces or commas.
Formula Used
This calculator estimates the dominant eigenvalue with power iteration. It then applies simple deflation to estimate the second eigenvalue and measure dominance strength.
- Power iteration step:
x(k+1) = A x(k) / ||A x(k)|| - Rayleigh quotient:
λ ≈ (xᵀAx) / (xᵀx) - Residual check:
r = ||Ax - λx|| - Dominance ratio:
|λ1| / |λ2| - Spectral gap:
|λ1| - |λ2|
How to Use This Calculator
- Enter a square matrix, one row per line.
- Provide an initial vector or keep the default values.
- Choose tolerance, maximum iterations, and normalization type.
- Click Calculate Dominance to estimate the dominant eigenpair.
- Review the dominance ratio, spectral gap, eigenvector, and convergence plot.
- Export the result as CSV or PDF when needed.
Example Data Table
| Case | Matrix | Approx. Dominant Eigenvalue | Approx. Second Eigenvalue | Dominance Comment |
|---|---|---|---|---|
| Example A | 5 1 0 1 3 1 0 1 2 |
5.732051 | 3.000000 | Moderate dominance with steady convergence. |
| Example B | 8 0 0 0 2 0 0 0 1 |
8.000000 | 2.000000 | Strong dominance with very fast convergence. |
| Example C | 3 1 1 3 |
4.000000 | 2.000000 | Clear dominance and stable power iteration behavior. |
Matrix Model Dominant Eigenvalue
In a matrix model, the dominant eigenvalue is the eigenvalue with the largest magnitude. It often controls long-run growth, decay, ranking strength, or steady-state behavior when the matrix is applied repeatedly.
FAQs
1) What is the dominant eigenvalue?
The dominant eigenvalue is the eigenvalue with the greatest absolute value. In repeated matrix multiplication, it usually governs the main growth pattern, decay rate, or long-run direction of the system.
2) What does matrix model dominant eigenvalue mean?
In a matrix model, the dominant eigenvalue describes the strongest long-run multiplier. It is commonly used in population models, ranking systems, Markov-style processes, and repeated transformation problems.
3) How does this calculator estimate the dominant eigenvalue?
It uses power iteration. The method repeatedly multiplies an initial vector by the matrix, normalizes the result, and tracks the Rayleigh quotient until the estimate becomes stable or the residual becomes small.
4) Why is the spectral gap important?
The spectral gap compares the largest and second-largest eigenvalue magnitudes. A bigger gap usually means stronger dominance and faster convergence of power iteration toward the leading eigenvector.
5) What does the dominance ratio tell me?
The dominance ratio is |λ1| / |λ2|. Larger values indicate that the dominant eigenvalue clearly outweighs nearby competitors, so iterative methods usually settle faster and the leading behavior is easier to interpret.
6) Why might the method fail to converge well?
Weak spectral separation, repeated eigenvalues, nearly defective matrices, poor starting vectors, or dominant complex behavior can slow convergence or create unstable estimates in real-valued power iteration.
7) Can repeated or complex eigenvalues affect the result?
Yes. Repeated eigenvalues can weaken dominance, and complex dominant pairs may cause oscillation. In those cases, the reported estimate can be approximate and may need a more specialized eigensolver.
8) How should I choose the initial vector?
Any nonzero vector usually works if it has a component in the dominant eigenvector direction. A simple all-ones vector is often enough, but a better informed guess can reduce iterations.