Questions tagged [matrix]

Matrix is a rectangular array of elements (e.q. numbers, symbols, or expressions), arranged in columns and rows.

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4
votes
2answers
97 views

Is there an efficient way to form this block matrix with numpy or scipy?

Is there an efficient way to form this block matrix with numpy or scipy? $$ \left[ \begin{array}{cccc} \mathbf{B} & \mathbf{0} & \cdots & \mathbf{0}\\ \mathbf{AB} & \mathbf{B} & \...
4
votes
0answers
76 views

An optimization method for bounding the eigenvalues of a unknown non symmetric matrix

Given a positive objective function $f$ that acts on a real-valued matrix $A$, I am interested in the following problem $$\underset{A \in \mathbb{R}^{n \times n}}{\text{minimize}} \quad f(A) \quad \...
3
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1answer
95 views

Matrix condition number and reordering

Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
2
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0answers
23 views

Minimize number of math operation of a specific matrix vector multiplication?

Let's say we have a Matrix M and a column vector v like below multiply equals Assume we can only perform multiplication, addition and substraction operation. With normal approach we need 3 ...
3
votes
4answers
5k views

cartesian products in numPy

Have two arrays, in my case $X = \{1,2,\dots, n\}$ or X = np.arange(n). How do I get $Y = X \times X = \{ [i,j]: 1 \leq i,j \leq n \}$ as a 2D array in numPy? ...
2
votes
1answer
34 views

numpy.outer without flatten

$x$ is an $N \times M$ matrix. $y$ is a $1 \times L$ vector. I want to return "outer product" between $x$ and $y$, let's call it $z$. z[n,m,l] = x[n,m] * y[l] ...
2
votes
0answers
35 views

Best way of porting code from the GPU to MPI-nodes

I have a program, structured in two parts, $A$ and $B$. Both parts are capable of running as standalone units, and written in C++. $A$ is written for cluster systems, running entirely on CPU-nodes, ...
0
votes
1answer
47 views

what does “D = diag(W.1)” means?

, what does “D = diag(W.1)” means?on page #2, just below equation (6) PFA screenshot and here is the link of the paper - original paper
4
votes
1answer
63 views

Fast algorithm for computing cofactor matrix

I wonder if there is a fast algorithm, say ($\mathcal O(n^3)$) for computing the cofactor matrix (or conjugate matrix) of an $N\times N$ square matrix. And yes, one could first compute its determinant ...
2
votes
1answer
124 views

A Bound for the inverse of the sum of identity and triangular matrix

I wonder if there are any theorems which can help me to calculate an upper bound for the spectral norm of: $$\left\| \left[ I + \sum_{i=1}^{\overline{n}\in\mathbb{N}} \big( C_i - I\big)\right]^{-1}\...
5
votes
1answer
368 views

Fast algorithm for computing matrix square root using randomized linear algebra?

Is there a fast algorithm for computing the matrix square root of a real symmetric matrix using random matrices or randomized algorithms?
2
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0answers
31 views

Randomized Submatrix of a Sparse Matrix

I have a sparse square matrix $A$ with size $n \times n$ and number of nonzero entries $nnz$. The goal is making a sub-matrix $B$ with $s$ nonzeros which are randomly chosen from $A$. Duplicates are ...
2
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2answers
4k views

Matlab element within ranges and distance between two points

I have a $53534\times 3$ matrix with $x$, $y$, and $z$ coordinates. I want to find the element of matrix within ranges as follows: ...
0
votes
1answer
42 views

Plotting ratings matrix

Hello fellows and folks. I have been looking to do this for 1 month and still cannot find the way to do it. Here’s what’s going on: I have a csv file called ratings.csv with the following ...
2
votes
1answer
100 views

BLAS operation question

I want to perform the following operation: $$ A = A + U B^T $$ where $A$ is $m \times n$ dense, $U$ is $m \times m$ upper triangular, and $B$ is $n \times m$ dense. The BLAS function ...
0
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1answer
40 views

Computing excited states using itensor (with DMRG)

I am trying to compute first few excited states of some Hamiltonian (I am using itensor and its DMRG algorithm). To do so, I am ...
4
votes
1answer
153 views

Accurate way of getting the square root inverse of a positive definite symmetric matrix

What is the most accurate algorithm to get the square root inverse of a positive definite symmetric matrix? I am not looking as much for efficiency, though using quadruple precision computation is out ...
5
votes
2answers
173 views

Computing any element of the null space of a singular matrix

Given a singular matrix $A$, what is the fastest method to find a single non-zero solution to $Ax=0$? Note that we are not looking for the whole kernel, we just want any non-zero vector in it. I ...
11
votes
3answers
807 views

Testing if two 12x12 matrices have the same determinant

I am given a $12 \times 12$ matrix $Q$ that is symmetric, invertible, positive definite and dense. I need to test if $$\det(Q) = \det(12I-Q-J) \; \; (1)$$ where $J$ is the all ones matrix. I am ...
4
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0answers
599 views

Perron-Frobenius theorem on general real symmetric matrices

From the Perron-Frobenius theorem, it might be concluded that the spectral radius is the largest eigenvalue for positive matrices, ie, matrices with strictly positive entries. In other words, the ...
2
votes
1answer
130 views

Converting matrices L and U output by dgssv() of SuperLU to triples format

How can I convert matrices L and U output by dgssvx() of SuperLU to triples format (to ...
5
votes
0answers
437 views

Sparse matrix format and sparse-matrix sparse-matrix multiplication

I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices). Mainly I have to multiply such ...
2
votes
1answer
22 views

Determine image of hypercube under linear map

Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb ...
2
votes
1answer
61 views

Rank of Hadamard Product with Masked Matrix

I have a matrix $A\in\{0,1\}^{d\times n}$ and $rank(A)=d,d<n$, and another matrix $X\in \mathbb{R}^{d\times n}$, but I do not know the rank of $X$. What can we say about the rank of their Hadamard ...
1
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2answers
40 views

denoting a variable as a matrix using octave syms package

I'd like to use the syms package to do some algebra for me, but the baseline assumption seems to be that variables are scalars. I would like to denote some variables as matrices. This will change the ...
3
votes
2answers
67 views

Convexity of Sum of $k$-smallest Eigenvalue

If I have a real positive definite matrix $A\in\mathbb{R}^{n\times n}$, and denote its eigenvalues as $\lambda_1\leq \lambda_2 \leq ... \leq \lambda_n $. Define the function as $f(A)=\sum_{i=1}^{k} \...
0
votes
0answers
54 views

Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking ...
0
votes
1answer
46 views

An Upper Bound of $\left<H,X\right>_F$ with Constrainted Rank

We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$...
5
votes
2answers
109 views

Choose a subset of $m$ columns that maximize $|A^T A|$?

I have a set of $n$-dimensional vectors, and would like to choose $m$ of them to become the columns of an $n\times m$ matrix. I would like to choose the subset that maximizes $|A^T A|$, where $A^T$ is ...
2
votes
1answer
54 views

Problem of multiplication of big (sparse) matrix with numpy (python)

I wanted to multiply two simple (big and sparse) matrix with numpy. And I saw that the calculation fails when matrices are too big. If i take $X$ a random vector (size $n$). With pandas, I ...
1
vote
1answer
134 views

How do I solve the matrix equality constrained optimization problem using Lagrangian multipliers?

Solve the following minimization problem in $\mathbf{X} \in \mathbb{R}^{m \times n}$ $$\begin{array}{ll} \text{minimize} & \frac 12 \| \mathbf{X}\mathbf{X}^T -\mathbf{A} \|^2_\mathcal{F}\\ \text{...
5
votes
1answer
143 views

How can I apply Euler's Method to predict a point in time rotating around multiple axis'

I am xposting this from my original stackoverflow question where I was presented with a coding challenge that I have been able to narrow down extensively and I think it lies with Euler's Method. Here'...
2
votes
1answer
104 views

Projection onto the set of Orthogonal matrices

Let $M \in \mathbb{R}^{n \times n}$ and denote the set of Orthogonal matrices by \begin{equation} \mathcal{O}_{n} = \left\lbrace Q \in \mathbb{R}^{n \times n} \colon QQ^{T} = \mathbb{I}_{n} \right\...
6
votes
1answer
234 views

Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
9
votes
3answers
199 views

What is the reason that LAPACK uses $\tau$ in QR decomposition (instead of normalizing the reflection vector)?

LAPACK's QR routine stores Q as Householder reflectors. It scales the reflection vector $v$ with $1/v_1$, so the first element of the result becomes $1$, so it doesn't have to be stored. And it stores ...
2
votes
1answer
420 views

How to convert MPIAIJ to SEQAIJ matrix in petsc/petsc4py?

I am curious, if there is a function to convert MPIAIJ (distributed matrices in AIJ format) to a SEQAIJ matrix that lie on a single processor. It is possible to do such an operation for PETSc vectors ...
4
votes
0answers
70 views

Block matrix and DSYRK

I want to compute the matrix $$ A = \sum_{i=1}^N v_i v_i^T $$ where each $v_i$ is a given vector of length $2500$, so that $A$ is $2500 \times 2500$, and my $N$ is about 2 million. Rather than call ...
16
votes
4answers
5k views

Row major versus Column major layout of matrices

In programming dense matrix computations, is there any reason to choose a row-major layout of the over the column-major layout? I know that depending on the layout of the matrix chosen, we need to ...
0
votes
2answers
313 views

closed form approximation of matrix inverse with special properties

I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties: invertible positive definite ...
3
votes
2answers
131 views

Parallel assembly of matrix

I have a matrix which I want to assembly quickly, which is in block form: $$ A = \pmatrix{ A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}} $$ ...
3
votes
1answer
152 views

How to find the nearest/a near positive definite from a given matrix?

I'm given a matrix. How do I find the nearest (or a near) positive definite from it? The matrix can have complex eigenvalues, not be symmetric, etc. However, all its entries are real valued. The ...
1
vote
0answers
71 views

Fast calculation of $A^T B$

I need to compute a matrix-matrix product, $A^T B$, where $A$ is $n \times r$ sparse, and $B$ is $n \times q$ dense. The number of rows $n$ is far larger than both $r$ and $q$. In fact $n$ is so large ...
-4
votes
1answer
89 views

what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
1
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2answers
61 views

Access optimized data structure for representing integer lattice

Consider the integer lattice in $2d$, namely the set $\mathbb{Z}^2 = \{(x,y): x,y\in \mathbb{Z}\}$, and let $u:\mathbb{Z}^2 \to \mathbb{R} $ be a function defined on some bounded subset of $\mathbb{Z}^...
1
vote
1answer
135 views

Solving linear system with matrix multiplication

When solving a linear system $Ax=b$ where $A=B^TCB$ do I need to form $A$ explicitly by two matrix-matrix multiplications or is there another more simple way? $C$ is a NxN matrix and not always ...
4
votes
1answer
154 views

Do I really need to invert this matrix

I need to calculate a matrix $A$ (at least some elements of it, see below) as defined by the following equation $$ A=B(\mathbb{1}-B)^{-1} $$ where B is a square matrix of dimension $N$ and $\mathbb{...
1
vote
0answers
71 views

finding null space to a complex matrix

I need to solve the following equation: $$ \begin{pmatrix} \frac{\omega^2}{c^2}\varepsilon_x-\mu_z^{-1}k_y^2-\mu_y^{-1}k_z^2 & \mu_z^{-1}k_xk_y & \mu_y^{-1}k_xk_z\\ \mu_z^{-1}k_xk_y &\...
8
votes
3answers
506 views

Rule of thumb for sparse vs dense matrix storage

Suppose I know the expected sparsity of a matrix (i.e. the number of non-zeros / total possible number of non-zeros). Is there a rule of thumb (perhaps approximate) for deciding whether to use sparse ...
3
votes
0answers
135 views

Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix

I want the eigenvalues of the following generalized eigenvalue problem: $$ Av = \lambda M v $$ where $A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite $M\in\mathbb{R}^{n\...
1
vote
1answer
59 views

Thomas Algorithm Kernel OpenCL

I am trying to implement the Thomas algorithm using OpenCL. ...