Questions tagged [matrix]
For questions about using and representing matrices on a computer in order to solve computational problems. Should generally also include a tag about the specific property/problem you are solving (e.g. [tag:linear-algebra], [tag:eigenvalues], [tag:inverse].
600
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Overlap matrix and its inverse matrix
Now, we consider a non-orthonormal basis:
$$\mathcal{S}_N=\{|\alpha\rangle,a^\dagger|\alpha\rangle,a^{\dagger 2}|\alpha\rangle,\ldots,a^{\dagger N}|\alpha\rangle\},$$
where $|\alpha\rangle$ is the ...
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Finding the Vector $v$ for a Given Householder Matrix Transformation of Non-Collinear Vectors $a$ and $b$
Consider a vector $v$ in $\mathbb{R}^{n\times1}$. The Householder matrix is defined as follows:
$$H(v)=I-\dfrac{2vv^T}{v^Tv}.$$
It can be demonstrated that $H(v)$ is symmetric and orthonormal. The ...
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Efficient Algorithm for LU-Factorization of Modified Matrix with Last Column Alteration If We Have Its Not-Modified LU-Factorization
Suppose that we have a $n\times n$ matrix $A$. We have its LU-factorization as $A=LU$ (or $PA=LU$ that $P$ is a permutation matrix). Now assume we change the last column of matrix $A$ and denote the ...
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Why is it that SVD routines on nearly square matrices run significantly faster than if the matrix was highly non square?
In Python / Matlab, if you run a routine for SVD on a significantly non-square matrix, X, such as X.shape = (2,15000) you will ...
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3
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1
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How can we calculate mixed derivatives numerically using the Chebyshev derivative matrix?
Using the Chebyshev derivative matrix $D$, we can numerically approximate the first and second derivative of a function by doing matrix multiplication:
$${df(x) \over dx} = Df(x) \tag 1$$
$${d^2f(x) \...
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2
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0
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Complex matrix logarithm discontinuity by solving inverse Fourier integral by alternative method to FFT
NOTE: This code is a piece of code I am using for a master's thesis, so I do not expect someone to do the work for me, but I gladly accept suggestions of any kind.
However, I am trying to get the ...
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numerical calculation of haldane model arm chair edge states
hello I am trying to numerical simulate the band structure of the one-dimensional periodic arm chair edge states, I use the pybinding model to construct and ...
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1
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0
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Sums of sparse matrices modulo 2
I have a set of sparse matrices modulo 2 (i.e systems of equations in the format "a XOR b XOR ... XOR c = 0 or 1" compressed by me into reduced row echelon form).
This is a rather large set (...
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2
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Reordering eigenvalues in Schur factorization - MATLAB orschur and LAPACK dtrsen not producing the same results
Disclaimer: I previously posted this on SO, but though it would be more relevant for scicomp. The original post has been deleted.
I have been trying to recreate the functionality provided by MATLABs <...
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84
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Tools to compare two matrices with same dimensions
Context:
I have two 3D non-random matrices that have the same dimensions. These matrices represent satellite images with 1 band, so their values are strictly positive. They both present areas that ...
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1
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Measuring the extent to which two sets of vectors span the same space
I have a set of measurements $y_i$, $1 \leq i \leq N$, and I want to model these measurements with a linear model. I have two possible models I can use,
$$
y \approx A c
$$
and
$$
y \approx B d
$$
...
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Calculating camera calibration matrix with Scilab
I'm not entirely sure whether this question belongs here or in DSP but I think this is the proper site.
I'm following these videos (first video, second video) to calibrate a camera for photogrammetry ...
- 131
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1
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Automatic differentiation (AD) of a loss function which maps unitary matrix onto number
Is it possible to estimate whether automatic differentiation (AD) techniques could enable a more efficient way to repeatedly compute the derivative $\delta L / \delta u^*_{ij}$ of a specific loss ...
- 111
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1
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Reshaping a matrix and rearranging the elements lexicographically into a vector
Let us say that I have a $3 \times 3$ matrix $\bf X$, that has to be reshaped into a vector and rearranged as follows:
$$ {\bf v} ({\bf X}) = \begin{bmatrix} x_{22} & (x_{23}+x_{32}) & (x_{21}+...
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When is Lanczos tridiagonalization accurate?
Suppose that we are given a random, symmetric matrix $A$, and a random vector $q$. For concreteness, assume the dimensions of $q$ and $A$ are both $1,000$. I would like to use the Lanczos algorithm to ...
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1
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QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift on ...
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2
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1
answer
198
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Parallelize pseudo inverse of a matrix using Lapacke
I am currently using the protocol described in
https://stackoverflow.com/questions/55599950/computation-of-pseidoinverse-with-svd-in-c-using-blas-and-lapacke to compute the pseudo inverse of a matrix.
...
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5
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632
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Matrix derivative
I am looking to compute the derivative of the following expression:
$$\frac{\partial}{\partial X}\mathrm{tr}\left[A\exp(X)\right]$$
where $A$ is both a symmetrical and positive-definite matrix and $X$ ...
- 436
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1
answer
106
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Scipy solve_ivp sensitivity to random phase shifts
I am trying to solve a coupled system of ODE's using the solve_ivp function from scipy. The general form of the equation is given via
$$\dot{y}(t) = M(t)y(t).$$
The time dependence of matrix is ...
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1
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153
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How to efficiently transpose distributed matrix in Scalapack?
I have a distributed matrix in block cyclic layout. Is there an efficient way to out/in place transpose a distributed matrix with scalapack?
Context: I am trying to diagonalize the transpose of a ...
- 103
3
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1
answer
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Rewriting quadratically-constrained optimization problem as a semidefinite program
Suppose $A,H$ are positive definite matrices and $\alpha,t$ are scalars. Is there a way to massage the following problem into a form suitable for a specialized solver?
$$\begin{array}{ll} \underset{\...
- 2,273
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Does cblas_dgemm mutate my input matrices?
I have written a matrix class Matrix<T> for which I have implemented a wrapper function for cblas_dgemm.
...
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3
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211
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How do we compute a matrix-vector product without using the standard matrix multiplication?
I am a bit confused right now. I am taking a class on numerical linear algebra and as I have understood sometimes one doesn't compute the Matrix-Vector product $Av$ normally but uses other techniques ...
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1
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111
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QR decomposition with only diagonal elements changing
I have to compute the QR decomposition of a matrix A repeatedly, each time with ONLY diagonal elements changing. Is there an efficient way to accomplish this ...
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3
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1
answer
227
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BiCGSTAB convergence
So I need a fast converging solver for SysLinEq as a subroutine in fortran, decided to test BiCGStab in Matlab.
Thank God I decided to test it out on first before implementing in Fortran as a ...
- 384
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1
answer
373
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Matrix regularisation for ill-conditioned problems
I have read that matrix regularisation can improve the stability of LU or Cholesky decomposition of ill conditioned problems. The idea is to add a value to the diagonals of a matrix:
$B=A+cI$
In the ...
- 840
2
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0
answers
52
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Is there a permutation used in sparse QR factorizations that better locates small elements on the diagonal of R?
Is there a permutation used in sparse QR factorizations that better locates small elements on the diagonal of R to the end of the diagonal? As an example, consider the following snippet of MATLAB ...
- 757
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359
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Recurrence relation for matrices
I have matrices ($S_0$ thought $S_N$) and I have a recurrence relation that link successive matrices together.
$$S_i + S_i(aS_{i-1}^{-1})S_i=C_i+aS_{i+1}$$
We can assume for this problem that $S_0=S_N=...
- 436
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45
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Lowest eigenvalues of a matrix with divergent entries
In high energy physics we oftentimes encounter the following problem. For a given parametrized matrix $\{H_{ij}(\Lambda)\}$, we know that in the limit $\Lambda\to\infty$ some of its entries become ...
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1
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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 ...
- 219
1
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1
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165
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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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Dynamic Sized Identity Matrix in Eigen
I am aware of creating an identity matrix in Eigen if the number of rows and columns are known. How do we create them dynamically when the size is not known? An example would be useful. Thanks.
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299
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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 ...
- 165
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1
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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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1
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Converting a 4 rank matrix to 2 rank matrix after using tensorproduct
Let's say I have a 2x2 matrix (with symbols) called 'A'. Now, if do
B = sympy.tensorproduct(A,A)
print(sympy.shape(B))
I get,
...
5
votes
1
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461
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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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2
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nnz-preserving sparse matrix multiplication
Let A and B be sparse matrices in $ \mathbb{R}^{m\times m}$ with (roughly) the same density $p$. I want to efficiently compute a matrix $C$ that in some sense is "closest" to $AB$ while ...
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Symmetric Matrix in Eigen C++
I am aware of a symmetric matrix type in uBLAS as ublas::symmetric_matrix matrix. Is there an equivalent for this in Eigen library that can be used to construct one or do we need to explicitly check ...
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0
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293
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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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Is this a legit way to sample a random matrix spectrum?
In order to undergird a theoretical model concerning many body physics, I want to have exponentially large eigenvalue spectra from the random matrix GOE ensemble. its properties are mainly
(i) a ...
0
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2
answers
133
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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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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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3
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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 ...
- 383
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1
answer
690
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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 ...
- 185
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1
answer
372
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2-norm and infinty norm of a system in controls
How to compute 2-norm or infinity norm of following system?
i am confused whether to calculate using simple matrix theory "where it don't regard for s domain" or H2 and H-infinty norm. ...
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1
answer
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What is difference between L2 norm and H2 Norm?
When someone refers 2-norm of system,L2 and H2 are used interchangeably by author and is rather confusing. Even the matlab has different functions for H-infinity norm and L-infinity norm.
as shown in ...
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0
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Can a symmetric matrix be found using matrix-vector operation while maintaining matrix-vector operatin?
Consider $A\in\mathbb{R}^{n\times n}$ its not a special matrix and in the worst case all of its entries are non-zero. I am looking for a way to compute $AA^{T}$ using matrix-vector operation. The ...
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163
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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 ...
user14831
1
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2
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689
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Dyadic operations, fourth order tensors and Tensor algebra
I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving ...
2
votes
1
answer
137
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Vectorization Matlab/Octave of Markov Matrix Powers
I have just created a code snippet in Octave/Matlab that aims to create a plot which shows the accuracy of an initial probability vector $\vec{\pi}$ derived from the transition probability matrix $\...
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