# Tagged Questions

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### Eigen values of L2 projection Matrix

If I have an arbitrary set of $L^2$ functions $\{\phi_i\}$, then want to find the projection onto the subspace of $L^2$ generated by the basis, i.e $span\{\phi_i\}$, I believe I just need to solve ...
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### First eigenvalue of the given linear operator

I have the following question: Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$. Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by $Lu = -Du'' + Cu$, where $D$ is a positive ...
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### Vandermonde question

I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem): I have ...
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### The value interpretation of eigenvectors.

My question is may be strange but I wanna lie it any way. The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
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### Characterizing (or deriving) the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$ ...
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This is an attempt to generalize the answer to a previous question Consider the $n \times n$ matrix $$A=\left[ \begin{array}{cccc} 0 & \frac{1}{n-1} & ... & \frac{1}{n-1} \\ 1 & 0 ... 0answers 91 views ### Biharmonic operator Consider the problem:$$ \Delta^2 u = f$$on the square domain U=(0,1)\times(0,1) with boundary conditions:$$ u(x,y)=\Delta u(x,y) = 0 for $(x,y) \in \partial U.$ I try to solve it with the ...
Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
I would like to ask a question about singular values of matrices of the form $A^TA$. We know that by Courant minimax principle the singular values are given by (in increasing order \$s_1 > ...