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9 votes
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Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
Kung Yao's user avatar
  • 192
7 votes
0 answers
195 views

Hölder continuity of spectrum of matrices

Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
Jochen Glueck's user avatar
2 votes
0 answers
537 views

Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices

recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
MarcPN's user avatar
  • 21
2 votes
0 answers
121 views

Eigenvalues of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
Chriscrosser's user avatar
2 votes
0 answers
279 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
BaoLing's user avatar
  • 329
1 vote
0 answers
62 views

What do you call this class of matrices with a unique positive eigenvalue associated to a graph?

I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
M. Winter's user avatar
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0 votes
0 answers
107 views

Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
Bazin's user avatar
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