# Tagged Questions

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### Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy$

Consider the operator $$Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy$$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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### Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
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### What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
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### Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
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### Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
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### Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
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### Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
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### eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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### Eigenvalues and fft of sound trying to find similarities

I'm looking at sound resonance patterns using matlab/octave to see if there may be patterns between FFT and Eigenvalues. I can get the frequencies and each of the frequencies amplitude to recreate ...
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### Removing extraneous solutions from an eigenvalue equation

I have an eigenvalue problem of the form $\left[ L_1 + \dfrac{L_2}{\Omega} + \dfrac{L_3}{\Omega^2} + \dfrac{\Omega-1}{\Omega+\eta}\right] \phi(x) = 0$ which I am trying to solve for the complex ...
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### Finding linear transform matrix from characteristic polynomial

I got two similiar very simple question on a notebook. 1)let characteristic polynomial $P_A(x)=x^2+2x-3$ and $T:V\to V$ and DimV=2,S={$\alpha_1,\alpha_2$} is ...
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### Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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### Eigenvalue of a linear transformation

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$.If $p \epsilon V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
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### Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
Quantum mechanics math question: Suppose that there is eigenstate $|q \rangle$ where $q$ is position observable . The question is, 1) What is eigenstate? How is this different from eigenvector? ...