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Consider a diagonalizable matrix A. For example, a real symmetric matrix. Start with any vector b and keep multiplying it with A. A A A ... A b The vector
by srean 8mo ago
Consider a diagonalizable matrix A. For example, a real symmetric matrix. Start with any vector b and keep multiplying it with A.
A A A ... A b
The vector that the result will converge to is a scaled version of one of the eigenvectors of the matrix A.
But which one ? The one with the largest eigenvalue among all eigenvectors not orthogonal to b.
https://en.wikipedia.org/wiki/Power_iteration https://en.wikipedia.org/wiki/Power_iteration
- Sharlin 8mo agoAh… that "diagonalizable" is doing some heavy lifting there! I was wondering how exactly you’re going to make, say, a rotation matrix to converge anything to anything that’s not already an eigenvector. And rotation matrices certainly aren’t degenerate! Though apparently non-diagonalizable matrices can be called defective which is such a dismissive term :( Poor rotation matrices, why are they dissed so?!
- srean 8mo agoLove them, those rotation matrices. Take logarithm of the eigenvalues and you get back the angle. This to me had solidified the notion that angles are essentially a logarithmic notion ... Made more rigorous by the notion of exponential maps