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Questions tagged [linear-algebra]

Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

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Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components

I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
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261 views

Markowitz Pivoting to reduce size of a dense integer system

I am dealing with a large sparse integer matrix that I need to find the nullspace of. I've seen Markowitz Pivoting come up in several places discussing similar problems such as here: http://www....
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148 views

Efficient way to do congruent transformation using matrix inverse?

I know a square self-adjoint matrix $S_{vv}$ and I want to find: $S_{rr} = HS_{vv}H^{\dagger}$ where $\dagger$ denotes conjugate transpose. I do not know $H$ but I do know $H^{-1}$. What is the ...
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995 views

Pixel-To-Angle Transformation in Camera Image

I'm trying to localize points I see in a camera image in terms of azimuth and elevation and match points between shots. Individual shots should differ only in rotation around the camera's center (...
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28 views

PCA performed on a configuration with scaled axes

Suppose a configuration $X\in\mathbb{R}^{n\times 2}$ is output of PCA on some high-dimensional data $Y\in\mathbb{R}^{n\times h}$. Note that this PCA is performed by $$X=Y\cdot U,$$ where columns of $U$...
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Numerical Range of a matrix in Python

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex $n\times n$ matrix $A$ is the set $$W(A)=\left\{{\frac {{\mathbf {x}}^{*}A{\...
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64 views

How to make a directed graph symmetric?

Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
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48 views

Formula for overdetermined logical matrix pseudoinverse not requiring SVD?

In https://commons.wikimedia.org/wiki/File:YI_%3D_PI.png, you will find a formula-based solution for an overdetermined logical matrix pseudoinverse. This simple formula gives the same result as the ...
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76 views

Ill-conditioned stiffness matrix

I am writting a Fem code in c++ for a 2d plane stress model. My question is regarding the assembly stiffness matrix.I noticed that some elements of the matrix are not exactly zero but insted a number ...
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47 views

Norm estimates if adjoints can't be computed

Assume that you have two linear maps $A$ and $V$. For a given $x$ (of appropriate dimension) you can compute $Ax$ numerically, and for any $y$ (of appropriate dimension) you can calculate $V^Ty$ ...
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71 views

Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
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123 views

A parallelized GMRES solver?

My application calls for solving a dense, 40,000 x 40,000, ill-conditioned linear system. The native SciPy GMRES solver with preconditioning has worked well for my application and solving a single ...
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89 views

Is there a subfield within computational science research that's done on pencil and paper?

Is there an area of computational science research that can be done on pencil and paper (with results written up for a journal format later on)? I'm wondering if there is abstract proof-based linear ...
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38 views

A preconditioner for self-consistent iteration

I tried to derive a preconditioner for self-consistent iteration similar to section IX in arXiv:0804.2583. For simplicity, consider here only one orbital (one or two electrons) systems. Suppose that ...
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315 views

Cholesky factorization of a block matrix

Let a Matrix $\mathbf{Y=[A\;b;b^T\;}d]$, where $d$ is a scalar, and $\mathbf{b}$ is a vector. How to find the cholesky factorization of $\mathbf{Y}$ if i know the cholesky factorization of $\mathbf{A}$...
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135 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for c/c++ implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I ...
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38 views

Damping vs. Weighting in large sparse linear inverse problems

For large sparse inverse problems, there are two terms that I don’t have a clear perception of them; **Weights and damping values (regularization parameter) **. How can we determine weights for ...
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55 views

Find mass matrix in a system of linear equations

Given $z_t=\sum_{i=1}^t \theta_iz_{t-i}+v_t $, where $t=1,...,N$ where $N=1024$. I need to write this in matrix form (a system of linear equations) as $\mathsf{A}\mathsf{z} = \mathsf{z} - \mathsf{v}$. ...

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