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Unrelated to the range of a matrix, but it strikes me how similar those plots of eigenvalues of named matrices are to pole/zero plots of named filters in signal
by colanderman 3y ago
Unrelated to the range of a matrix, but it strikes me how similar those plots of eigenvalues of named matrices are to pole/zero plots of named filters in signal theory. I wonder where the correspondence lies.
- icegreentea2 3y agoThe eigenvalues of a filter are exactly the poles.
- tomrod 3y agoEigenvalues show how much a matrix stretches an input vector in the direction of a corresponding eigenvalue. I really, really love the intuition matrix analysis gives about the world!
- patrec 3y agoCorresponding eigenvector, you mean, right?
- tomrod 3y agoDuly corrected. Yes, corresponding eigenvector!
- patrec 3y agoBTW, what are some favorite examples of yours for matrix analysis derived intuition about the world?
- tomrod 3y agoSo, let's take one example, SVD - Mathematical foundations for principal component analysis - Compression algorithms by removing lower-value singular values - Original design for PageRank - Kabsch algorithm / Wahba's problem - Literal rocket science! - Quantum information Another one that blows my mind is the Gershgorin disk/circle theorem to bound the spectrum of a matrix. Matrix analysis was where I fell in love with mathematics and it ceased to be a constant churn of copy pattern-do problem-repeat and I started to see the structures behind the curtain.
- patrec 3y agoThank you!
- colanderman 3y agoAh, a filter in state-space representation: https://www.dsprelated.com/freebooks/filters/State_Space_Filters.html https://www.dsprelated.com/freebooks/filters/State_Space_Fil...