Questions tagged [numerical-linear-algebra]

Questions on the various algorithms used in linear algebra computations (matrix computations).

2,233 questions
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Book for manifolds that are used in scientific computation

I've been reading up on papers in scientific computation that utilize terms such as "Stiefel manifold" and "Grassmannian" that seem to be invoked in numerical linear algebra. I don't have any ...
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Numerical Analysis is giving me some trouble. In specific, I'm highly confused by our definition of numerical stability. I'm hoping some of you can help me clear my confusion. Definition. An ...
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Is there any relation between SVDs of two matrices with same range?

Let $A$ and $B$ be two symmetric and positive semidefinite matrices with the same size. Further, assume that $A$ and $B$ share the same column space (i.e., $\mathcal R (A) = \mathcal R (B)$ ). Is ...
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It is known that the solution to the linear system $Ax=b$ where $A$ is symmetric and positive definite is the minimizer of the quadratic function $f(x)= \frac{1}{2} x^T A x - x^T b$ We can solve it ...
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Variety of submodules of a finitely presented module

Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1\to F_0\to M\to 0$. I want to undertand the collection of finitely ...
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Householder transformations to upper triangular form

Let $A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$. How to transform this matrix with Householder transformations to an upper ...
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Efficiency in matrix multiplication with diagonalizable matrix. [closed]

Are there efficient algorithms to compute Ax in the case of A being diagonalizable instead of not being?
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Convergence of CG method, number of eigenvalues

I am trying to fix this proof of a lemma that I did't correctly write: If $A\in \mathbb{R}^{n\times n}$ symmetric positive definite with $m$ different Eigenvalues $\lambda_j$, then the Conjugate ...
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Can we simplify $A^{-1}Bx = x$ where $A$ is a block matrix with each block being diagonal and half the blocks of $B$ are zero?

I have the following eigenvalue problem involving block matrices $A$ and $B$: $$A^{-1}Bx = x. \quad \quad \quad \quad (*)$$ $A$ and $B$ have special structures. I would like to reduce/simplify this ...
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Is it possible to find complex eigenvalues with QR decomposition?

I wonder if it's possible to find the complex eigenvalues with QR decomposition. I can find the real eigenvalues with QR just by doing this. ...
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Orthogonalize two sets of vectors against one another

Given two subspaces $U,V\subset \mathbb{R}^n$ and orthonomal bases $U = \{u_1,\ldots,u_p\}$ and $V = \{v_1,\ldots, v_q\}$ (wlog $p\geq q$) is there a fast way to compute an orthonormal basis for $U+V$?...
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Bring ODE into a suitable form to solve it with Runge-Kutta steps

Can anyone please help me understand, how I should bring this ODE y'' + y = sin(t) with initial conditions y(0) = 100, y'(0) = 5 into a Runge-Kutta-Form? I tried to solve this equation, the ...
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What is the best algorithm to find the inverse of matrix $A$

I'm going to build a C-library so I can do linear algebra at embedded systems. It's most for machine learning. https://github.com/DanielMartensson/EmbeddedAlgebra/ Anyway, I need to compute inverse ...
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Why does the Lanczos algorithm fail on repeated/multiple eigenvalues?

I'm trying to code up the Lanczos algorithm for eigenvalue approximation at the moment. I've seen on pages like this that the algorithm can't distinguish the eigenvectors if the dimension of the ...
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Generating a random sparse hermitian matrix in Python

I'd like to find a way to generate random sparse hermitian matrices in Python, but don't really know how to do so efficiently. How would I go about doing this? Obviously, there are slow, ugly ways to ...
Construct $A =Q\Lambda Q^T$. $Q$ is found by applying $QR$ factorization to $B=$randn($n$), and $\Lambda$ is defined to be \begin{align*} \Lambda = \mathrm{diag}(\lambda_1,\lambda_2,\ldots,\...
I am trying to create a direct proof that if the matrix A is self-adjoint and positive-definite, then the Gauss-Seidel iteration converges for any initial ${\bf x}_{0}$ I think I need to prove \$\rho({...