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Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?

The $n\times n$ imaginary matrix $A$ satisfies $A^\top=-A$, so it is skew-symmetric. The Youla decomposition is $$A=iO\Sigma O^\top,$$ where $O$ is a real orthogonal matrix and $\Sigma$ is a real ...
Carlo Beenakker's user avatar
4 votes

First eigenvalue of the spherical cap

For the 2-sphere $\mathbb{S}^2$, the first eigenvalue of the spherical cap can be calculated via stereographic projection. Under this projection, $U(r)$ is a ball $B_{\mathbb{C}}(0,\tan(r/2))$ in $\...
Boey C's user avatar
  • 41
2 votes

Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

Dan Popovici deforms the exterior derivative operator on a complex manifold to $D_{\eta}=\eta \partial+\bar\partial$ for a positive function $\eta$. He was apparently led to this construction by ...
Ben McKay's user avatar
  • 25.7k
3 votes

Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

Witten already discusses deformations for other complexes including the signature complex in his first paper. He formulated Morse inequalities for the Dolbeault complex on K"ahler manifolds in a ...
Gibbon's user avatar
  • 51
4 votes

Spectral theory: a key to unlocking efficient insights in network datasets

A place to find a wealth of examples is in graph signal processing, where spectral decomposition of the Laplacian is known as the graph Fourier transform. A. Ortega, Introduction to Graph Signal ...
Georg Essl's user avatar
5 votes

Book on Hilbert spaces, including non-separable

Halmos's Introduction to Hilbert space and the theory of spectral multiplicity is what you are looking for. Since it's published by Dover, the price is very reasonable.
Robert Furber's user avatar

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