All Questions
Tagged with perturbation-theory matrices
91
questions
0
votes
2
answers
37
views
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
0
votes
0
answers
24
views
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
2
votes
1
answer
50
views
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
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
0
votes
0
answers
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
1
vote
0
answers
23
views
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
6
votes
0
answers
206
views
Eigenvectors not stable under matrix perturbation
Let $A,B$ be square matrices of the same size but arbitrary components. It is a fact that the eigenvalues of $A$ and $A+\epsilon B$ are close for $\epsilon$ small, but the eigenvectors may be ...
- 6,360
1
vote
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
6
votes
1
answer
317
views
Eigenvalues of $\left[\begin{matrix}1&\epsilon&2\epsilon\\\epsilon&1&2\epsilon\\-2\epsilon&-2\epsilon&2\end{matrix} \right]$ as a series in $\epsilon$
Consider the matrix
$$
M := \left[ \begin{matrix} 1 & \epsilon & 2\epsilon \\ \epsilon & 1 & 2\epsilon \\ -2\epsilon & - 2\epsilon & 2 \end{matrix} \right]
$$
for some small ...
- 2,767
2
votes
0
answers
68
views
Linear Algebra Question and Matrix Perturbation Problem Solution
I found the following formulation of a perturbation problem in my research (see attached image). In step 4c to 4d, the equation goes from:
$$|I+A^{-1}e_k\Gamma^T| = 0$$
$$|1+\Gamma^TA^{-1}e_k| = 0$$
...
- 21
1
vote
0
answers
82
views
Perturbed Gram Matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first cannonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(...
- 477
1
vote
1
answer
464
views
Inverse of symmetric positive definite perturbation of symmetric positive definite matrix
Let $A$ be an $n\times n$ real invertible matrix, and $\delta A$ be a $n\times n$ matrix such that $A+\delta A$ is invertible. Then, it is known that
$$
\frac{\left\|(A+\delta A)^{-1} - A^{-1}\right\|}...
- 1,450
0
votes
1
answer
250
views
Shifting a matrix
Let $n\in \mathbb{N}$, and $A\in \mathbb{R}^{n\times n}$ be a semi-positive definite matrix. What can we say about the matrix
$$ A_h:= A+(h-1)I, $$
where $I$ is the identity matrix, and $h>0$ (...
- 187
2
votes
1
answer
806
views
Angle and distance between subspaces.
Let $\text{Sin}(\Theta)$ be the diagonal matrix containing the $Sine$'s of principal angles between two subspaces $\mathcal{A}$ and $\mathcal{B}$, and let $\Pi_A$ and $\Pi_B$ be their respective ...
- 104
2
votes
1
answer
509
views
Eigenvalue perturbation of hermitian matrices
For $\rho$ and $A$ hermitian matrices, $t$ a scalar parameter, $f_i(\rho)$ denoting the $i-$th eigenvalue of the matrix $\rho$ and $|x_i\rangle$ the normalized eigenvector of $\rho$ corresponding to $...
- 181