How to use the Power Method Eigenvalue Calculator
- 1. Enter the matrixOne row per line, up to 8×8. Optionally give a starting vector such as 1 0 0.
- 2. Choose the eigenvalueLargest |λ| uses A; smallest |λ| applies the method to A⁻¹.
- 3. Read the tableEach row shows A·x, the largest entry (the λ estimate) and the scaled vector.
How the power method works
Start with a vector x₀ that is not zero. Multiply by A, then divide the result by its entry of largest size, so the new vector has largest entry 1. That divisor is the current estimate of the dominant eigenvalue λ₁, and the scaled vector approaches its eigenvector. Repeat until both stop changing.
It works because every vector is a mix of eigenvectors, and multiplying by A stretches the λ₁ part the most. The error shrinks like (λ₂/λ₁)ᵏ, so convergence is fast when |λ₁| is well ahead of |λ₂| and slow when they are close. For the smallest eigenvalue, apply the method to A⁻¹: its dominant eigenvalue is 1/λ_min.
Worked example
Take A = [[2, 0, 1], [0, 2, 0], [1, 0, 2]] and x₀ = (1, 0, 0). A·x₀ = (2, 0, 1), so λ ≈ 2 and x₁ = (1, 0, 0.5). A·x₁ = (2.5, 0, 2), so λ ≈ 2.5 and x₂ = (1, 0, 0.8). A·x₂ = (2.8, 0, 2.6), so λ ≈ 2.8 and x₃ = (1, 0, 0.929).
The estimates go 2, 2.5, 2.8, 2.93, 2.98 … and reach λ = 3 with eigenvector (1, 0, 1). The other eigenvalues are 2 and 1, so the ratio 2/3 sets the speed.
Good to know
If the starting vector has no component along the dominant eigenvector, choose another one. For all eigenvalues of a 2×2 or 3×3 matrix at once, use the eigenvalue and eigenvector calculator.
Frequently asked questions
Why divide by the largest entry?
It keeps the numbers from growing or shrinking without bound, and that largest entry converges to the dominant eigenvalue.
What if two eigenvalues have the same size?
If λ₁ = −λ₂ or they are complex, the estimates oscillate and do not settle. The tool then stops at the maximum number of iterations.
How is the smallest eigenvalue found?
The tool inverts A and runs the power method on A⁻¹. The smallest eigenvalue of A is 1 divided by the result.
How can I check the answer?
The tool shows |A x − λ x|. A small value means x really is an eigenvector for λ.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.