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MathidiaMathematics Academy
Scientific computing
Lesson 57 of 75
Easy practice

Computational eigenproblems

Objective. Use power and QR iteration ideas.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Computational eigenproblems approximate eigenvalues and eigenvectors of large matrices without forming the characteristic polynomial, using iterative methods such as power iteration and the QR algorithm. Power iteration repeatedly multiplies a vector by A and normalises it, converging to the eigenvector of the dominant (largest magnitude) eigenvalue provided there is a gap between it and the next eigenvalue. For example, starting from v₀ = (1,0) and A = [[3,0],[0,1]], Av₀ = (3,0), which normalises to (1,0), already the dominant eigenvector for eigenvalue 3. The QR algorithm instead repeatedly factorises A = QR and forms A' = RQ, converging towards a triangular matrix revealing all eigenvalues.

Worked example

For computational eigenproblems using power iteration on A = [[4,0],[0,1]] starting from v₀ = (1,1), find the normalised direction after one iteration (largest entry scaled to 1).

4 … 0
Illustrated card showing 4 … 0
  1. 1Compute Av₀ = [[4,0],[0,1]](1,1) = (4,1).
  2. 2Identify the largest magnitude entry of (4,1), which is 4.
  3. 3Divide each entry by 4 to normalise: (4/4, 1/4) = (1, 0.25).
  4. 4This vector (1, 0.25) is the approximate dominant eigenvector direction after one iteration.

Answer: (1, 0.25)

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Common mistake. A common mistake is applying power iteration expecting convergence to the dominant eigenvalue when the two largest eigenvalues are equal in magnitude, since without a strict dominance gap the iterates do not converge to a single eigenvector.

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