# Questions tagged [perturbation]

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### Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix. For real $t$, let us consider the one-parameter family $$H(t) = H + t V$$ of Hermitian matrices. Kato's perturbation theory ...
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### Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...
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Consider a non-linear operator $\cal H$ which maps a function to a function (e.g., a map from a starting wave function $f(x,y,z)$ to a later wave function according to some non-linear PDE) and an $\... 1 vote 0 answers 208 views ### componentwise eigenvector perturbation Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple. If this is trivial, I ... 2 votes 1 answer 687 views ### Null Space Perturbations Hi, I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases. The distilled version of the ... 6 votes 2 answers 654 views ### analytic approximation of a non-negative matrix by a sequence of positive matrices Let$L \in \{0,1\}^{n \times n}$be a non-negative matrix whose row sum is 1. ($L$stands for "limit"). It is known that there exists a unique$r \times r$principal minor of$L$that is a permutation ... 0 votes 1 answer 192 views ### Regular Perturbation Series soln to eqn I want to find the a 3 term perturbation soln of (i)$(1+x)^3 = ex$where$e\ll1$Direct substitution of the regular perturbation series$x = x_0 + ex_1 + e^2x_2$into (i) does not work I ... 3 votes 2 answers 499 views ### Number of perturbations of the Jordan form I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. For example, if a Jordan form consists of a single cell 2x2 $$J=\... 4 votes 1 answer 415 views ### What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants? To each closed 3-manifold N, there is a corresponding Witten--Reshetikhin--Turaev invariant Z_k(N) depending on an integer k (the level) and a Lie group G (and perhaps we'll just concentrate ... 3 votes 1 answer 320 views ### enlarge the separation between two matrices The separation between two square matrices A and B, often used as a measure of the sensitivity of invariant subspace problems, is defined as$$ \operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\... 10 votes 3 answers 12k views ### Kind of submultiplicativity of the Frobenius norm:$\|AB\|_F \leq \|A\|_2\|B\|_F$? Let$\|\cdot\|_F$and$\|\cdot\|_2$be the Frobenius norm and the spectral norm, respectively. I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from BIT ... 3 votes 0 answers 440 views ### A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values) Let$A$,$B$be$n\times n$Hermitian matrices. Suppose$\alpha_1\geq \alpha_2 \geq \...
Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...
I am learning perturbation theory and would like to be able to determine where boundary layers are going to occur just by looking at the differential equation. Let $n\in\mathbb{N}$ and $p_i(x)$, \$0\...