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8 votes
3 answers
547 views

Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)

We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...
Sharpie's user avatar
  • 183
5 votes
4 answers
2k views

Differentiability of eigenvalue and eigenvector on the non-simple case

Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
Shake Baby's user avatar
  • 1,638
4 votes
2 answers
477 views

Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true. There exists universal ...
Alex Wenxin Xu's user avatar
1 vote
1 answer
77 views

Efficient way to compute eigenvalue decomposition for following problem

I have an optimization problem $$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I \end{array}$$ where $A\in R^{n \times n}$ and it is symmetric positive definite, ...
jason's user avatar
  • 111
1 vote
0 answers
183 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
knuth's user avatar
  • 33
1 vote
0 answers
67 views

Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
THC's user avatar
  • 4,547
1 vote
0 answers
158 views

Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph. Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
Val K's user avatar
  • 355
0 votes
0 answers
74 views

Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
  • 4,058