What is the numerical range of a matrix?

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42 points·jjgreen··13 comments

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.

The eigenvalues of a filter are exactly the poles.

Ah, a filter in state-space representation: https://www.dsprelated.com/freebooks/filters/State_Space_Fil...

Eigenvalues 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!

Corresponding eigenvector, you mean, right?

Duly corrected. Yes, corresponding eigenvector!

BTW, what are some favorite examples of yours for matrix analysis derived intuition about the world?

So, 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.

Thank you!

Note that this concept can extended to linear operators, which are the generalization of matrixes. Since it’s a generalization, some results for matrixes (finite dimensional operators) don’t hold for infinite dimensional operators.

Related notion to numerical range is numerical shadow. And we have an entire website about it:

https://numericalshadow.org/

Applications?

Avoiding this https://youtube.com/watch?v=vfl33Tn0pYc

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