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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3 votes
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Column-normalized inverse?

Suppose we define $A^{*}$ of positive definite $A=X'X$ using following two steps: let $B=A^{-1}$ scale columns of $B$ to obtain a matrix with $1$'s on the diagonal For the case of singular $A$, we ...
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7 votes
0 answers
56 views

Choice between using Moore-Penrose inverse and G2 inverse

Moore-Penrose inverse for an arbitrary matrix $X\in \mathbb{R}^{n \times p}$ is defined by a matrix $X^\dagger$ satisfying all of the Moore-Penrose conditions, namely \begin{align} (1) \;\;\;& XX^\...
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  • 173
2 votes
1 answer
114 views

Numerical representation of linear spaces

A linear space in mathematics is a set whose elements (vectors) have operations of addition and multiplication by a scalar defined in such a way that certain properties are satisfied (commutativity, ...
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9 votes
2 answers
2k views

Is there an algorithm or graph theory that allows me to not need to store an intermediate matrix when calculating AT*Y1*A + BT*Y2*B?

I have a system of conductors for which there are two dense matrices of the (complex) mutual admittances, $Y_A$ and $Y_B$, which are symmetric. Then, an equivalent nodal admittance matrix $Y_N$ is ...
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3 votes
1 answer
83 views

Using submatrices of matrix decomposition for solving a large number least-squares problems

I want to decrease the computational time for solving a large number (>1000) of least-squares problems. Given a matrix, the system matrix for each least-squares problem is a submatrix of the given ...
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  • 119
1 vote
1 answer
94 views

constructing a symmetric matrix for finite difference

I come across the following operator in a paper $\mathcal{I}\psi = \psi_{xxxx} + (r~\psi_x)_x$, where $\psi=\psi(x)$ and $r=r(x)$. Periodic boundary condition is employed. It claims that the operator $...
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  • 217
0 votes
0 answers
37 views

Solving Symmetric/Hollow Matrix issue

I have a particular issue and need something creative or solution from calculus. I have Symmetric/Hollow Matrix, a numbers are % of mismatch between them. Ideally, all of them should be 0, but I have ...
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0 votes
0 answers
71 views

How to estimate stability and stiffness of a system of coupled ODEs?

I'm running into issues with Python/Julia ODE solvers requiring prohibitively small timesteps to evolve a system of 4 coupled ODEs (the order of magnitude of the state variables and time unit span ~40-...
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5 votes
1 answer
100 views

Largest singular value without using the adjoint

The square of the largest singular value of a linear map $A$ can be computed by using the power iteration for $A^TA$ and one advantage of this is that the iteration is matrix free, i.e. you only need ...
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  • 1,728
1 vote
2 answers
64 views

Calculate average distance between pairs of points without computing full distance matrix

Suppose I have a set of $N$ points of shape $N \times D$, where $D$ is the dimensionality. I want to compute the average Euclidean distance between all points, as well as additional moments such as ...
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  • 165
2 votes
1 answer
92 views

Finding the correct order of eigenvectors of a parameter-dependent Hermitian matrix

so, I have a symmetric, analytic matrix $\mathbf{H}(x)$ ($x$ is real). Because $\mathbf{H}(x)$ is analytic and $x$ is real, it is possible to find analytic functions for the eigenvectors and the ...
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  • 153
-1 votes
1 answer
38 views

Finding matrix of a linear transformation using R programing

The full question is: Let {u1, u2,···, un} and {x1, x2,···, xm} be bases for Rn and Rm respectively. Let T:Rn→Rm be the linear transformation whose associated matrix with respect to the above bases is ...
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2 votes
1 answer
103 views

Numerical diagonalization of Hamiltonian

Framework I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian. How I tried ...
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  • 21
3 votes
1 answer
63 views

Requesting for Finite Difference Methods reference in Portuguese or English

Crossposted on Mathematics SE I have been assigned a group project for an introductory Linear Algebra subject on Finite Difference Methods and sparse matrices. Our professor advised we use Gilbert ...
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  • 33
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0 answers
37 views

Matrix 3D rotation/translation/scaling relative to each other

I am trying to implement an algorithm, that removes a part of an assembly (3D) and translates/rotates/scales those other matrices which are child/parent to a new matrix. For example: (MT = ...
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  • 101
2 votes
0 answers
42 views

Solving linear system and obtaining operator norm

I need to solve a linear system of the form $(\mathrm{Id} + \mathbf{J})\mathbf{x} = \mathbf{b}$ for $\mathbf{x}$ and I also need to compute the operator norm of $\mathbf{J}$ (i.e. the largest singular ...
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1 vote
1 answer
75 views

Fast algorithm to compute chi-square

I would like to evaluate the chi-square of the form $\chi^2=v^{T}C^{-1}v$ where $v$ is a column vector and $C$ is a covariance matrix. Both $v$ and $C$ are known and $C$ is a $740\times740$ matrix. ...
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  • 123
0 votes
0 answers
170 views

Solving huge dense square symmetric linear system

I have a linear system of the type $A x = y$ where A is a dense, square, symmetric, positive definite matrix, $x$ a vector of unknown parameters, and $y$ is a vector of observed quantity. I know that ...
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  • 121
5 votes
1 answer
175 views

Converting distance matrix back into original data

Suppose that we have $N$ points, and a distance matrix $D \in \mathbb{R}^{N \times N}$ describing the Euclidean distance among those points. For now, assume that we do not necessarily know how many ...
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  • 165
6 votes
1 answer
113 views

Fast way of computing the action of a matrix power on a vector

For integer $k>0$, it is well-known that one can use binary exponentiation to evaluate the matrix power $\mathbf A^k$, where $\mathbf A$ is an $n\times n$ matrix. However, it is not clear to me if ...
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7 votes
0 answers
101 views

Can we sparse solve a few eigenvalues specified by index range?

I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
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9 votes
2 answers
233 views

Efficiently estimating trace of a product of matrices

I have $d\times d$ real-valued matrices $A_1,\ldots,A_k$, $1000<d<4000$, $k\approx 50$, and need to estimate the trace of the following matrix product $$t=\text{tr}(A_1 A_2\cdots A_k A_k^T \...
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0 votes
0 answers
84 views

Weighted Jacobi Not Working on 1D Poisson (Issue with Optimal $\omega$)

I've been trying to learn some numerical linear algebra, and I decided to try to implement the weighted Jacobi method to the 1D Poisson problem $$-u''(x)=f(x),\qquad u(0)=a,\ u(1)=b,$$ where we ...
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5 votes
0 answers
118 views

Why are fast Givens rotations mentioned so little in the recent literature?

Fast Givens rotations seem like a nice upgrade from standard Givens rotations. They take fewer multiplications to apply and avoid the calculation of a square root. They are, however, given very little ...
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2 votes
2 answers
68 views

How to find fundamental matrix based on other fundamental matrix and camera movement?

I am trying to speed up some multi-camera system that relies on calculation of fundamental matrices between each camera pair. Please notice the following is pseudocode. ...
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  • 121
0 votes
1 answer
50 views

why is $Hx= x-2u^{H}xu$ ? why not $Hx= x(x-uu^H) + 2((u^Hx)u)^2? $

I have some confusion in this diagram My confusion : why is $Hx= x-2u^{H}xu$ ? why not $Hx=x(x-uu^H) + 2((u^Hx)u)^2? $ My thinking is that by using pythogoras theorem blue line(vector) denotes ...
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  • 101
0 votes
1 answer
99 views

Incomplete Cholesky factorization algorithm

I want to implement incomplete Cholesky factorization to precondition, the algorithm I refer from incomplete Cholesky factorization, ...
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1 vote
0 answers
50 views

Trouble inverting complex matrix with numpy and scipy

I have some matrix-valued, complex data $Z(f)$ with $f\in\{f_0,f_1,\dots\}$ and $Z(f_i)$ being a 3x3 matrix. I require the inverse $Z^{-1}(f)$ in my workflow. After encountering some problems with my ...
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4 votes
0 answers
74 views

Decomposing a banded matrix

Suppose we have a linear algebra problem with a banded matrix A which has nonzero entries on the main diagonal, two nearest sub-diagonals, and two other sub-diagonals (such band structure often arises ...
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10 votes
1 answer
481 views

Generalization of eigendecomposition problem

Let $A\in \mathbb{R}^{n\times n}$ and $v \in \mathbb{R}^n$. We recognize $Av=\lambda v$ for some scalar $\lambda$ as an eigendecomposition problem. Suppose $\mu \in \mathbb{R}^n$, and let $\odot$ ...
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  • 173
4 votes
1 answer
216 views

Is there any way/any python function to calculate the condition number of the roots of a polynomial directly?

I know that NumPy has linalg.cond(A) to find the condition number of a matrix A. But, if I want to find the condition numbers of the roots of a large polynomial ...
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  • 41
2 votes
1 answer
95 views

Can someone explain the equivalence between Oja's rule and PCA in a simple way?

I have to give a presentation on unsupervised learning in 2 days, and I have to explain/show the equivalence between Hebb's learning rule (or Oja's rule to be more specific) and PCA. The thing is that ...
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2 votes
0 answers
97 views

Regularisation of ill-conditioned matrix-vector problem

I have a linear* problem which arises from an integro-differential system, and writes: $$ (\mathbf{I}+\lambda \mathbf{A})x = b $$ where $\mathbf{A}$ is a real full matrix, size $n\times n$, but is not ...
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6 votes
1 answer
229 views

PETSc-like library for Julia

I want to build an application for Material Point Method (and probably other meshfree methods too) in Julia and I am looking for library for direct and iterative solvers that can help me with it. One ...
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5 votes
2 answers
480 views

Computational method to compute both the (log) determinant and inverse of a matrix

Suppose I have a square matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ and a vector $\mathbf{b}\in\mathbb{R}^n$. In my application I need to accomplish two things. I need to find the solution of the ...
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4 votes
1 answer
108 views

Solving for a single element of a solution of a linear system

I wish to solve a linear system $A x =b$ in which $A$ is dense but not too large, say no larger than $10\times10$. However, I am not interested in the full solution vector $x = [x_0, x_1, \dots]$, ...
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  • 695
2 votes
0 answers
66 views

Multigrid method: linear solver and modified residual

I am trying to better understand the FAS multigrid algorithm for Euler equation in FV discretization. The usage of the modified residual (the residual with forcing) inside the different cases: ...
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  • 382
3 votes
1 answer
88 views

Find $x$ that satisfy $(I-A^*A)+x(\frac{A+A^*}{2})\prec0$ using LMI or SDP on Matlab

Given $A\in\mathbb{C}^{n\times n}$, I want to use LMI or SDP to find feasibility of $x>0$ in the following inequality: $$(I-A^*A)+x(\frac{A+A^*}{2})\prec0,$$ where $D\prec0$ means that $D$ is ...
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  • 183
4 votes
0 answers
125 views

Stable iterative solver for complex symmetric linear systems

I am interested in the iterative solution (preferably Krylov-type solvers) of a problem $\boldsymbol{A}x=b$, with $x,b\in\mathbb{C}^{n\times1}$ and $\boldsymbol{A}\in\mathbb{C}^{n\times n}$. $\...
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  • 141
0 votes
2 answers
127 views

Implementation of $[X, \cdot]$ as an $n^2 \times n^2$ matrix, where $X$ is an $n \times n$ matrix

Let $M_n(\mathbb{R})$ denote the set of $n\times n$ matrices with real entries. I have an $n\times n$ matrix $X\in M_n(\mathbb{R})$, and I would like to implement the linear operator $[X, \cdot] : M_n(...
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5 votes
2 answers
115 views

Optimizing a quadratic form integral over unit sphere

I have an optimization problem, which is to maximize the following integral over the unit sphere: $$ \max_B \int d\Omega \mathbf{f}^{\dagger}(\theta,\phi) (B^{\dagger} + B) \mathbf{f}(\theta,\phi) $$ ...
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0 votes
0 answers
61 views

Comparing minimas of two different functions

The goal is to find vectors $x_u$ and $y_i$, both of the same length $f=64$, and to do this the following loss function is minimized: $$\sum_{u, i} (1 + \alpha \cdot r_{ui})(p_{ui} - x_{u}^{T}y_i)^2$$ ...
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6 votes
1 answer
198 views

Algorithm for solving systems which are nearly symmetric/adjoint?

I am familiar with Cholesky decomposition and LU factorization for solving systems of linear equations. I have a problem where I have large sparse matrices (say, 1000x1000 or larger) where only one or ...
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17 votes
3 answers
3k views

Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
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5 votes
1 answer
140 views

Solving absolute value systems

Let $z, b \in \mathbb R^n$, $A \in M_n (\mathbb R)$ and $|z| := (|z_1|, \dots, |z_n|)$. I am searching for an efficient algorithm to solve the absolute value system: \begin{equation} z - A |z| = b. \...
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1 vote
1 answer
237 views

Efficiency of scipy.sparse.linalg.expm_multiply with sparse vs unsparse vectors

From the package scipy.sparse.linalg in Python, calling expm_multiply(X, v) allows you to compute the vector ...
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1 vote
0 answers
87 views

Matlab - Equality between 2 Fisher matrices constructed in a different way

I want to know if, on a Fisher matrix, the projection operation (with a Jacobian matrix) commutes with a matricial inversion operation. The 2 ways to build these 2 matrices are: 1) First method: 1.1) ...
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0 votes
0 answers
60 views

PLASMA usage (Linear algebra routines that supports multithreading)

I have been looking for linear algebra libraries that support multithreading. I have found PLASMA which looks promising. It is from the same group that developed LAPACK. http://icl.cs.utk.edu/...
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0 votes
0 answers
145 views

Do the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and reinversion) commute?

I try to check the equality or the inequality between 2 Fisher matrices. The goal is too see if the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and ...
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3 votes
1 answer
196 views

How can I extract the banded or block diagonal part of a sparse matrix in MATLAB?

Given a large sparse (square) matrix in MATLAB, how can I extract the banded or the block-diagonal parts (of fixed size) of it efficiently? These are useful operations when prototyping and testing ...
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