7
votes
1answer
182 views

Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
0
votes
0answers
32 views

Eigenvalues of the linear operator $T^*T$

Let $T$ be a linear operator $T: H_1 \mapsto H_2$, where $H_1$ and $H_2$ are both Hilbert Spaces. Suppose further that $T$ is bounded, but not self adjoint. Suppose I also know that for functions ...
0
votes
0answers
23 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
1
vote
0answers
35 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
1
vote
2answers
48 views

Eigenvalues and eigenvectors of an operator

I have $Ku(t)=\int_0^1 G(t,s) u(s) ds, u\in L^2[0,1]$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ ...
0
votes
1answer
138 views

Zeros diagonal element of a semidefinite matrix leads to zeros row/column. Why?

I have a similar problem as in this question. In short words: Assume a square, positive semidefinite matrix $A\in\mathbb R^{n\times n}$. Show that if a diagonal element of $A$ is zeros then the ...
1
vote
2answers
106 views

Do the Fourier series of a function-valued Hermitian matrix converge?

Let $\mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a rank deficient periodic function-valued positive semi-definite Hermitian matrix. The entries $a_{ij}(t)$ of $\mathbf{A}(t)$ are ...
5
votes
1answer
120 views

Prove that an eigenvector is the maximum of a symmetric matrix

Let $f : S^{n-1} \rightarrow \mathbb{R}, x \mapsto x^TAx$ ( A is a symmetric matrix), then an eigenvector $\xi$ of A is a local maximum of this function. We are supposed to prove this in 6 steps and ...
0
votes
0answers
170 views

How does determinant relate to argument of matrix function?

I have a matrix function $R \mapsto J(R)$ from $\mathbb{R}$ to the set of irreducible matrices with non-negative entries. We can assume that $J(R)$ is $d \times d$, although any solutions that work ...
0
votes
0answers
133 views

Has this Principal Component Analysis (PCA) been done correctly?

I have a set of 3D data points, indicated by the blue color in the picture below. I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...
3
votes
1answer
50 views

Distinguishing between the different eigenvalues

Consider the symmetric matrix $$A=\begin{pmatrix} 2 & t & \cos t-1 \\ t & 2 & 0 \\ \cos t-1 & 0 & 2 \end{pmatrix}. $$ The (real) eigenvalues of $A$ can be found easily using ...
1
vote
1answer
79 views

Boundary-value problem in differential equations

Consider the problem: $$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$ Find the eigenvalues. How should one proceed about this problem? I am complete ...
1
vote
0answers
125 views

Would this be bounded?

Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of $M$ is less than $1$. Let $I_{r}$ be an $m$ ...
1
vote
0answers
185 views

Is there a great book on eigenvalues?

I keep encountering ostensibly very different branches of mathematics, only to have eigenvalues show up in each one. Is there a single book out there that presents a deep, unified account of the ...
0
votes
1answer
93 views

Eigenfunction of $(a(x) f^{II})^{II}= - \lambda^2f$

I need the eigenfunctions $f$ and eigenvalues $\lambda$ of $(a(x) f^{II}(x))^{II}= - \lambda^2f$ for a given $a(x)$. For $a(x)$ constant the solution is a combination of sin, cos, sinh and cosh. ...