All Questions
Tagged with perturbation-theory linear-algebra
115
questions
0
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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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0
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12
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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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24
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Find the condition number of a normal matrices.
Find the condition number of a normal matrices.
My attempt:-
I know condition number of $X\in \mathbb C^{n,n}$ is defined by $\kappa(X)=||X|| \cdot ||X^{-1}||.$ Definition of Normal matrix is given by ...
- 849
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35
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On estimating $\exp(-iHt)$ when $H$ is perturbed
Let us assume that we have an unbounded Hamiltonian $H$ and we perturb it a bit to be $H'=H+ \varepsilon A$. I am sure that estimating $||\exp(-iHt) - \exp(-iH't)||$ belongs to the subject of ...
- 634
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31
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Invertibility of the product of matrices when the norm is less than 1
I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the ...
- 25
1
vote
1
answer
115
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Given a perturbation to a symmetric matrix with multiple zero eigenvalues, seeking perturbation to eigenvalues
Let $A$ be a real, symmetric $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ where at least two of the eigenvalues are zero. Let $V$ be a real, symmetric $n \times n$ ...
- 420
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38
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Perturbation with positive diagonal
Suppose I have some matrix $A$ and I perturb it by some small amount: $A \to A + \delta A$. I am interested in determining the sign of the quantity $a^\dagger \delta A a$. Here is what I know:
$a$ is ...
- 101
2
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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
2
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38
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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 $\delta A$ ...
- 2,572
3
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2
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292
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Is Frobenius-norm projection Lipschitz continuous under operator norm?
Definition
Let $d \in \mathbb{N}$. Let $A \in \mathbb{R}^{d\times d}$ be a PSD matrix. Define the following operator: $T:\mathbb{R}_{+} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d} $: Let ...
- 714
1
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answers
25
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Bound on elementwise condition number in Higham
In chapter 7 of the book "Accuracy and Stability of Numerical Algorithms" by Higham there is an upper bound for the relative error of solution of linear system in terms of elementwise error. ...
- 51
1
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158
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Error bounds for perturbation of linear system input
I'm trying to determine an upper bound for the relative error when solving a linear system with a perturbed input. Particularly, I consider a linear system of the form:
\begin{equation}Ax = b\end{...
- 127
1
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0
answers
39
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Prove that this requirement can be met if and only if $B \in \mathcal{B}$.
(a) Let $R(t, \omega)$ denote the rotation matrix
$$
R(t, \omega)=\left[\begin{array}{cc}
\cos (\omega t) & -\sin (\omega t) \\
\sin (\omega t) & \cos (\omega t)
\end{array}\right] .
$$
Find ...
- 9,098
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40
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Bounds on $\|A(A^{\dagger}-B^{\dagger})\|$ in terms of $\|A^{\dagger}\|$ and $\|A-B\|$
Exponeitiation by `$\dagger$' indicates the Moore-Penrose pseudo-inverse and the norms are the usual matrix norm induced by the Euclidean norm. Let $rank(A)\leq rank(B)$. I'm wondering whether one can ...
0
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2
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218
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Can a small perturbation of a diagonal matrix increase its smallest eigenvalue to any arbitrarily large value?
Let $S\in\mathbb{R}^{n\times n}$ be a diagonal positive semidefinite matrix with exactly $k$ positive entries in its diagonal, where $k<n$. Let $\epsilon$ be any arbitrarily small positive real ...
- 19