# 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.

830 questions
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### Computing Algebraic Riccati inequality

In my Robust model control, I have got a couple of quadratic Riccati inequalities which need to be solved numerically on MATLAB. My question, Is there function on MATLAB can solve quadratic Riccati ...
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### Efficiently removing projection to subspace without having an orthogonal basis

I have a number of vectors $v_1, …, v_n$ and another vector $w$, that all are linearly independent, but not orthogonal. Let $V := \mathop{\text{span}}(v_1, …, v_n)$. I need to remove $w$’s projection ...
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### QR via Householder: less computationally complex variants?

I'm a probabilist and need to do a few computations for a rather big linear least squares problem, so I'm trying to optimize the computation as far as is feasible to me. In computing the QR ...
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### Role of non-hermitian coefficient matrices in the discretization of self-adjoint operators

What is the role of non-symmetric coefficient matrices in the solution of partial differential equations with self-adjoint operators? Particularly, I'm thinking about time-propagation of a linear ...
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### Closed-form Jacobian of se3 element w.r.t. 6-dof motion

Let $A$ and $B$ be two rigid transformations in 3D space that transform things from global to local coordinates. Let their relative transformation be expressed by $W=A*B^{-1}$. $W$ can also be ...
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### Eigen - Solving Complex Generalized Eigenvalue Problem

I've been using the Eigen C++ linear algebra library to solve various eigenvalue problems with complex matrices. I've recently had to use a generalized eigenvalue solution process, only to be ...
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### Updating factorization of Laplacian (add/remove edges)

For a graph $G=(V,E)$, recall that the unweighted Laplacian is $L:=D^\top D$, where $D\in\{-1,0,1\}^{|E|\times|V|}$ is the graph "gradient" operator that subtracts adjacent vertex values onto edges. ...
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Let $\Sigma$ be positive definite and the $D_i$ positive diagonal. Let the $X_i$ be unknown square matrices. Consider the system of equations: $$(I+\Sigma D_i)X_i=\Sigma\hspace{5mm}\text{for}\... 1answer 188 views ### Solving two inverse problems with same solution I've got two inverse problems,$$A_1 ~ x = b_1 \qquad A_2 ~ x = b_2$$So far I've been solving them independently using Tikhonov Regularization and getting two estimates for x. However in my case ... 1answer 366 views ### Rectangular Cholesky Decomposition Assume we have a positive semidefinite matrix A with A\in\mathbb{R^{n\times n}}. Clearly a cholesky decomposition A=B^TB exist with B\in\mathbb{R^{n\times n}}. For my research it would be ... 1answer 176 views ### Open Source Linear Algebra Library I am making a code in C that requires the equivalent of Matlab's '\' command for a linear system of the form AX=B where A is an NxN matrix and X, B are Nx1 vectors- i.e a code that performs X=A\B that ... 2answers 459 views ### Python environments for AMG and Gauss Seidel as solvers instead of preconditioners I am working on block preconditioning and seemingly it is common to write customised Krylov solvers for them. Within each solver, the individual block linear system with preconditioners are ... 0answers 97 views ### Eigenvalue decomposition of the sum: AA^T + diag(u) Suppose A\in\mathbb{R}^{n\times c},u\in\mathbb{R}^n,n\gg c. The time complexity of eigenvalue decomposing directly for matrix AA^T+\text{diag}(u) is O(n^3). And it is easy to avoid O(n^3) ... 0answers 30 views ### Difference between Chebyshev first and second degree iterative methods Consider linear equation Au = f. We want to solve it with iterative method (assuming A is good). First order iterative method is:$$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$The second degree ... 2answers 145 views ### Lanczos algorithms for Hermitian system with Toeplitz kernel Basically, I am trying to compute the SVD of a large Hermitian matrix H using Lanczos iteration, while H consists if a Toeplitz kernel K, which should be able to help speed up the matrix-vector ... 3answers 1k views ### Understanding Finite-Element Modal Analysis I am teaching a basic course on computational physics and for the last part of the course I will introduce freshman physics undergraduates to finite-element modelling methods. I am preparing a COMSOL ... 1answer 124 views ### How to determine the truncation error with products and quotients If I have an equation given by$$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$and I expand a,b,c,d in a Taylor series, where a is truncated at the A^{th} order, b is ... 0answers 128 views ### Rank-one Update to a Rank Revealing QR (RRQR) Factorization? Suppose we are given an RRQR factorization for some matrix A \in \mathbb{R}^{m \times n}, A\Pi = QR where m > n. Is there a cheap way to update A' = A + uv^{\top} given this factorization?... 3answers 167 views ### Eigenvectors of Black-box matrix \DeclareMathOperator{\diag}{diag} Consider the generalized eigenproblem A\mathbf{x}=\lambda B\mathbf{x}. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ... 2answers 516 views ### Zero Eigenvalues in Lanczos Algorithm I need to find the smallest few eigenvalues of a Hamiltonian (exact diagonalization) I use Python, and SciPy's built-in sparse eigenvalue solver. I notice, however, that for my small system (only a ... 1answer 66 views ### Show the symmetric Gauss-Seidel converges for any x_0 Let A\in\mathbb{R}^{n\times n} is symmetric positive definite and consider solving linear system Ax = b. Show that the symmetric Gauss-Seidel iteration converges for any x_0. Solution - Since ... 1answer 74 views ### Linear stationary iteration method Suppose you are attempting to solve Ax = b using linear stationary iteration method defined by$$x_k = G x_{k-1} + f that is consistent with $Ax = b$, i.e., for which $f = (I - G)A^{-1}b$. Suppose ...
Recall that a unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ is a lower triangular matrix with diagonal elements $e_i^{T}L e_i = \lambda_{ii} = 1$. An elementary unit lower triangular ...
Consider $A\in\mathbb{R}^{n\times n}$ whose nonzero elements are restricted to the main diagonal the strict upper triangular part, and the first subdiagonal. For $n = 8$ the locations that must be ...