I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i, …

When we use Green's function concept to solve a differential equation, for example to solve $(L+U)\psi=S$ we use the Green's funation $(L+U)G=\delta $ We can bring some part of le …

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence $$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\ …

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 ha …

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 $L$ t …

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-lin …

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 v …

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) d …

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 thi …

Given the integral: $$I(s)=\int_{-a}^a e^{s cos(t)}dt$$ is it possible to find an expansion of $I(s)$ using the Watson's lemma? Thanks in advance.

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 si …

Hi, Consider the first order autonomous ode for y(t) a scalar function of one real variable, t: (i) dy/dt = f(y,c) where c is some real parameter When 2 equilibrium curves (in t …

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 …

Let $||\cdot||_F$ and $||\cdot||_2$ be the Frobenius norm and the spectral norm. I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from …

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,...,\lambda_n$ Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that o …