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3 votes
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Maximal eigenvalue of a real symmetric Toeplitz matrix

The limit $\lim_{n \to \infty} \frac{\rho(A)}{n^2}$ is equal to $2/y^2$, where $y$ is the smallest positive real solution to $$2\cos y + 2 = y\sin y.$$ Indeed, in the limit, an eigenvector $\phi : [0,...
mathworker21's user avatar
  • 1,062
1 vote
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An example of a matrix whose eigenvalues fullfill 'No-resonance' condition

The matrix $$ \pmatrix{1&2\cr1&1\cr} $$ has eigenvalues $1+\sqrt2$ and $1-\sqrt2$, neither of which is an integer multiple of the other.
Gerry Myerson's user avatar
3 votes
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Function of eigenvalues of Laplacian matrix

Since the number of edges $e(G)$ equals the trace of Laplacian matrix of $G$ divided by 2, we have $$f(\mu_1,\mu_2,\dots\mu_n) = \frac{\mu_1+\mu_2+\cdots+\mu_n}2.$$
Max Alekseyev's user avatar
4 votes
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Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real

Let $A=(a_{ij})$ be a lower-triangular matrix with elenments $a_{ii}=\lambda_i$ (all distinct), $a_{n,n-1}=1$, $a_{i1}$ with $i>1$ arbitrary, all others are zeroes. Let $E=(e_{ij})$ have only two ...
Ilya Bogdanov's user avatar

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