Articles of numerical linear algebra

Algorithm to determine matrix equivalence

I’m a physicist who’s not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I’m just unaware of. I have two real rectangular matrices $A_{mxn} B_{mxn},$ where $m>n$ whose entries are $\pm1,0$. As a concrete example, take $$ A = \begin{bmatrix} 1 & 0 & 0 \\ […]

On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$. $$ \int_{\triangle}|\nabla u|^2=\gamma_{AB}(u(A)-u(B))^2+ \gamma_{AC}(u(A)-u(C))^2+\gamma_{BC}(u(B)-u(C))^2, $$ where $$ \gamma_{AB}=\frac{1}{2}\cot(\angle C), \gamma_{AC}=\frac{1}{2}\cot(\angle B), \gamma_{BC}=\frac{1}{2}\cot(\angle A). $$ What is a good reference for the formula? Is it due to R. Duffin? […]

Wiedemann for solving sparse linear equation

I am new member. I am researching in Wiedemann algorithm to find solution $x$ of $$Ax=b$$ Firstly, I will show a Wiedemann’s deterministic algorithm (Algorithm 2 in paper Compute $A^ib$ for $i=0..2n-1$; n is szie of matrix A Set k=0 and $g_0(z)=1$ Set $u_{k+1}$ to be $k+1$st unit vector Extract from the result of step […]

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on simple manifolds (e.g. on a unit sphere) shows that computed eigenvalues of $D$ often have non-zero imaginary parts, even though they should all be real. (For simplicity […]

How to prove $I-BA$ is invertible

This question already has an answer here: $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate] 3 answers

Strictly diagonal matrix

Suppose that matrix $A$ is strictly diagonally dominant, show that $$\|A^{-1}\|_{\infty}\leq\left[\min_i\left(|a_{ii}|-\left|\sum_{\substack{j\neq i}} a_{ij}\right|\right)\right]^{-1}.$$

Underdetermined linear systems least squares

I have an underdetermined linear system, with 3 equations and four unknows. I also know an initial guess for these 4 unknows. The article I am reading says: We can solve the system using the least squares method, starting form a guess. I don’t know how can I do this. Thanks.

What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?

Consider a full rank $n\times n$ symmetric matrix $A$ (coming from a set of physical measurements). I do an eigendecomposition of this matrix as $$A = E V E^T$$ Most of the eigenvalues are positive, while a few are negative but with much smaller magnitude compared to the maximum eigenvalue. I want to convert this […]

Minimum eigenvalue and singular value of a square matrix

How to show that the relationship $\left | \lambda_{min} \right | \geq \sigma_{min}$ holds between the minimum eigenvalue and singular value of a square matrix $A \in \mathbb{C}^{n \times n}$?

QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$ is an identity matrix, both of order $n$, and $\xi \in \mathbb{R}$ is a scalar. Is there an efficient way to compute a QR factorization […]