The "eigenvectors" of the derivative operator are the exponential functionspic.twitter.com/6KD6quJmEt
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We know that matrices can have eigenvectors, but it's more general than that.
Any 𝒍𝒊𝒏𝒆𝒂𝒓 𝒐𝒑𝒆𝒓𝒂𝒕𝒐𝒓 can have eigenvectors
The derivative is a linear operator
We call the eigenvectors of a differential operator 𝒆𝒊𝒈𝒆𝒏𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔pic.twitter.com/gKEbEL2Nez
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you can think about them like that.. technically the matrix and vector would have an uncountably infinite number of components.. i'll add to the thread about that!
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Perhaps the historically and physically most important case: The Fourier spectrum is the eigenvalue spectrum of a translationally invariant operator/kernel.
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Relates space and time translation invariance at a primitive level to the sinusoid. Also implies, among other useful things, that the Fourier spectral components of a linear stationary Gaussian process are statistically independent.
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Hermitian.
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Technically you’re talking about square n x n matrices. For rectangular m x n matrices, we get singular values, using singular value decomposition (SVD).
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Or a system of particles in qm canhave eugen values , relevant to the energu functional aka hamiltonian
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This is just fem, namely Ritz,where coefficients of trial wf ( variational methods) are but eigenvalue of discretized hamiltonian
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