# Questions tagged [eigenvalues]

eigenvalues of matrices or operators

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### Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real

Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...
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### Derivatives of a rotation tensor intended to rotate bases of a symmetric 2nd order tensor [closed]

Let $A = \sum_i \lambda_i N_i \otimes N_i$ and $B= \sum_j \mu_j n_j \otimes n_j$ , be the two symmetric and independent 2nd order tensors and their respective spectral decompositions. Let R be the ...
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### The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
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### Perron-Frobenius theorem to positive delay differential equations

The Perron-Frobenius theorem is that the largest eigenvalue (in modulus) of a non-negative matrix is real (and simple) and corresponds to a non-negative eigenvector. It is applicable to the positive ...
1 vote
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### Eigenvectors and eigenvalues of a symmetric matrix and its entry-wise absolute value

The modulus of matrices is meant componentwise in the following. Let $H$ be a sqaure matrix that satisfies the following assumptions: $H$ is real-valued, symmetric, and positive-definite.. $H$ is ...
1 vote
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### How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?

Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
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