All Questions
Tagged with perturbation-theory eigenvalues-eigenvectors
91
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How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix
If
D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
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Eigenvalue Spectrum of a non-Hermitian Matrix by a Hermitian Matrix Perturbation
Let $H_{eff}$ be a $n$ dimensional matrix defined by the eigenvalue spectrum $\Lambda$:
$$\Lambda(H)_n=\Lambda(H_n+H_{eff}),$$
Where $H$ is a infinite dimensional matrix, $\Lambda(H)_n$ are its lowest ...
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2
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37
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Sensitive eigenvectors to small perturbations in the matrix?
I've encountered a mathematical issue in my research. To provide some context, I have a known density matrix that I am reconstructing numerically using quantum data. The rebuilt matrix has ...
- 129
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Eigenvalue/eigenvector sensitivity in multidimensional scaling
From classical multidimensional scaling, a Cartesian coordinate matrix can be obtained as $\mathbf{X} = \mathbf{V} \mathbf{\Lambda}^{1/2}$, where $\mathbf{\Lambda}$ is a diagonal matrix of eigenvalues ...
- 528
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A case problem about rank-1-perturbation of diagonal matrices
I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it).
Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
- 181
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40
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Convergence rate of eigenvectors for perturbed matrices
Let $f(s)= (f_{1}(s),\ldots,f_{d}(s)), s \in \mathbb{R}^d, \textbf{o} \le s \le \textbf{1}$, be vector of probability generating functions, were each entry has finite third moments. Consider matrix of ...
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24
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
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109
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Derivative of Spectral Radius of Matrix $\exp(A(t))$
I am faced with the practical problem of solving a system
$$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$.
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- 569
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144
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Applying WKB Method for a Fourth Order Schrodinger Like Equation
I was trying to apply the WKB method to analyze the approximate eigenvalue condition for the Schrodinger like equation
\begin{equation}
\varepsilon^{4} y^{(4)} = (E-V(x))y, \quad y(\pm \infty) = 0, V(...
- 13
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38
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Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.
Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
- 78
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Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator
Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of ...
- 78
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37
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Leading eigenpair of degenerate non-symmetric matrix
Consider a non-symmetric matrix ${\bf A}_0$ with one eigenvalue $\lambda$ and all other eigenvalues are zero. The corresponding left $\bf u$ and right $\bf v$ eigenvectors to the eigenvalue $\lambda$ ...
- 135
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Finding the minimum eigenvalue of $A+itB\in \mathbb{C}^{n\times n}$
Suppose $A$ and $B$ are $n\times n$ real square matrices and further assume that $A$ and $B$ are symmetric. And now let $\lambda_A$ be the smallest eigenvalue of $A$.
Question : I want to know how $\...
- 2,908
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195
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Eigenvalues after Diagonal Perturbation
I was looking into how perturbation changes eigenvalues of a given symmetric matrix, and I came across lot of results regarding off-diagonal perturbation. But how does diagonal perturbation affect ...
- 256
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Given the eigendecomposition of a positive semidefinite, singular matrix $A$, how to determine if a small perturbation $A + \delta A$ is indefinite
Given the eigendecomposition of a positive semidefinite, singular matrix $A$, is there a good way to determine if a small perturbation $A + \delta A$ is indefinite? Here, the perturbation $\delta A$ ...
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