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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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3answers
338 views

Inverse of ill-conditioned symmetric matrix

I've got a matrix K, with dimensions $(n, n)$ where each element is computed using the following equation: $$K_{i, j} = \exp(-\alpha t_i^2 -\gamma(t_i - t_j)^2 - \...
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1answer
48 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 ...
4
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1answer
1k views

What is wrong with this matrix multiplication?

I am attempting to write a matrix multiplication routine because I need to do some analysis in CUDA and I want to validate it with CPU code. I am trying to use <...
4
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1answer
1k views

Nullspace algorithm for a sparse matrix

I am dealing with large, sparse, rational matrices that I need to determine the nullspace of. Currently, I have one that is about 12000x12000 (but not square), where one in every 2000ish elements is ...
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2answers
1k views

Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$. Many ...
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2answers
2k views

Algorithm for Sparse-Matrix Inverse

I have a $50000\times 50000$ matrix $A$ sparse matrix containing only 5 non-zero elements in each row. Now the problem is that the diagonal elements and the constants (in $B$ matrix such that $AX=B$) ...
4
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1answer
1k views

How to do transpose for trtrs (or tptrs) in blas?

How to do transpose for trtrs (or tptrs) in blas? I want to solve: XA = B But it seems that trtrs only lets me solve: ...
4
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1answer
316 views

$\mathbf{UDU}^\top$ decomposition routines in LAPACK/Eigen?

I would like to compute the decomposition of a real symmetric positive definite matrix $\mathbf{A} = \mathbf{UDU}^\top$. LINPACK seems to have it as DSIFA, but I ...
4
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1answer
286 views

Sparse LU for block-sparse matrices

I frequently need to solve linear systems with sparse matrices of moderate dimension (say a few thousand). These matrices are composed entirely of small dense blocks (typically 5-10 in dimension), and ...
4
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2answers
78 views

Any reference which summarizes decompositions?

Is there any reference (preferably available online as PDF, Free would be best) which summarizes the various matrix decomposition with their conditions for use, usage, algorithm, complexity and ...
4
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2answers
281 views

Is it possible to prove that the off-diagonal blocks of the Cauchy matrix have numerical rank $O(\log n)$?

Suppose we have a $n\times n$ Cauchy matrix of which the $ij$-th entry is given by: $$ A_{ij} = \frac{1}{a_i - b_j} $$ the assumption is that the distance between $\{a_i\}$ and $\{b_j\}$ is greater ...
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1answer
85 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 ...
4
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2answers
742 views

Smart way to multiply 3 matrices

I have a quantum mechanics simulation where I need to multiply three matrices that look like this: $$\rho(t_1)=U^\dagger \rho(t_0) \, U$$ where $U^\dagger$ is the hermitian conjugate of $U$. This ...
4
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1answer
664 views

How to improve this double-shift QR algorithm for non-symmetric matrices?

I've implemented a version of the double-shift QR algorithm featured in this report from ETH Zurich (Begins on page 77). The algorithm takes advantage of the Implicit Q theorem by applying an ...
4
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1answer
183 views

Efficient algorithm for a matrix product

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 ...
4
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1answer
131 views

Efficient way to generate a list of possible matrices (all integer components) with a determinant $V$

I have an interesting problem from my research that I have been struggling to solve. One part of the problem involves generating all possible matrices, where each set contains three integer vectors, ...
4
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1answer
417 views

Obtaining column vectors of pseudo-inverse of a matrix

I need to compute the pseudo-inverse of a very large rectangular dense matrix without any special structure or properties. I run out of memory/computing power and have no access to a large parallel ...
4
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1answer
93 views

Is there an efficient $O(n^2)$ way to get the eigen decomposition given a LDL factorization?

Let's say I have a LDL factorization of a matrix A. Is there an efficient $O(n^2)$ way to get the eigen decomposition of A given it's LDL factorization? Is there a more efficient way, in case L and ...
4
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1answer
160 views

Updating an approximate solution to a linear system in response to a small change

This question was original posted on SO but it was suggested that I post it here. I'm working on a program in which I have a banded matrix M and a vector b, and I want to maintain an approximate ...
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2answers
830 views

How can you calculate percent error in tensor approximations?

I have a matrix A which is an approximation to the known matrix B. Both matrices are square, 3x3 matrices and, in this case, are ...
4
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1answer
53 views

Configuration shift for determination of a true dimensionality

What would then be the way to determine a true dimensionality of a configuration of points $X\in\mathbb{R}^{n\times k}$ based on its Gram matrix $G=XX^T$? The "true" dimensionality refers to the ...
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3answers
218 views

Efficiently computing a few localized eigenvectors

Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$. The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
4
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1answer
281 views

What is the fastest method to invert millions of matrices?

My project involves large simulation and estimation. For each simulation I need to solve 600,000 systems of nonlinear equations. Currently I am using Newton's method to find the solutions. That ...
4
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1answer
157 views

Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of least-...
4
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0answers
90 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 \...
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0answers
73 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 ...
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108 views

How to stochastically estimate the trace of a matrix?

Specifically, the diagonal elements (can possibly both positive and negative) of the matrix can be computed efficiently but the total number is large ($\mathcal O(10^{18})$). My first thought about ...
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341 views

Optimisation of matrix exponential

I have a 7000x7000 sparse matrix (scipy), which I want to exponentiate. I've tried using scipy.sparse.linalg.expm, which works quite well for smaller matrices (takes a few seconds for a 1000x1000 ...
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199 views

Alternative to (costly) matrix multiplication

Consider the integral $$2\pi T = -\frac{1}{2} \int_0^{2\pi} A\frac{\partial B}{\partial \xi} \ d\xi$$ where $$A= \sum_{-N}^N i \ sign(n) \ B_n e^{-in\xi}; \quad \quad B= \sum_{-N}^N \ B_n e^{-in\xi}...
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142 views

How big a matrix can we row reduce in reasonable time?

I have very large matrices that I would like to row reduce (I need to keep track of the steps and find a basis of the kernel/image, not just find the rank). The good news is that I work mod 2 and the ...
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98 views

Tracking for two meshes

I'm dealing with physical simulation (position based dynamic). Now I'm trying to realize tracking for two meshes. In order to explain what does it mean, let's assume that we have two similar meshes: ...
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0answers
678 views

Generating pseudo-random orthonormal bases for random projection

I am performing series of random projections i.e. projecting the input matrix onto randomly generated orthonormal bases (of much lower dimensionality). The projection is just a matrix multiplication ...
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622 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 ...
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0answers
134 views

Probabilistic algorithms for matrix approximation

Considering regular matrix approximation inequality || $A - QQ^TA $|| < e where we try to approximate matrix $A$ by a lower rank orthonormal matrix $Q$. I've read an article on probabilistic ...
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3answers
2k views

Fast vector - “diagonal” matrix multiplication

Let $\mathbf{1}\in\mathbb{R}^d$ be a vector with all elements equal to $1$. Define: $$\mathbf{D} = \mathrm{diag}(\mathbf{1}^\top,\mathbf{1}^\top,\ldots,\mathbf{1}^\top) = \begin{bmatrix} 1 \cdots 1 &...
3
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2answers
245 views

Positive semi-definiteness of a (symmetric) matrix

Suppose a matrix $A\in\mathbb{R}^{n\times n}$ is given. Faced with a proof for $$x^TAx>0,$$ for a non-zero vector $x\in\mathbb{R}^{n}$, I was thinking to use the information of the spectrum of $A$ (...
3
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2answers
96 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} \...
3
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1answer
551 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 ...
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2answers
1k views

Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

I have many 3x3 real symmetric matrices for which I need to determine the eigenvalues. Wikipedia gives a nice non-iterative algorithm for this case, which I have translated into C++: ...
3
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3answers
675 views

Large overdetermined system of linear equations

I'm looking for a method to solve a large overdetermined system of linear equations in a least squares sense. The matrix is dense. I'd like to use a method that works even with limited memory (we can'...
3
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4answers
6k 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? ...
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1answer
111 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?
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2answers
2k views

BLAS, LAPACK or ATLAS for Matrix Multiplication in C

I am trying to find the most optimized way to perform Matrix Multiplication of very large sizes in C language and under Windows 7 or Ubuntu 14.04. And searching led me to BLAS, LAPACK and ATLAS. ...
3
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2answers
2k views

Matlab preconditioned conjugate gradient on big matrix

I have a sparse $5\,656\,236 * 5\,656\,236$ matrix $A$ with $166\,526\,888$ non-zero elements. The matrix comes from using the finite element method on a linear elasticity problem and is positive semi-...
3
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3answers
394 views

Evaluating large determinants with multivariate polynomial entries

I have some large (n~100) square matrices with entries two variable polynomials of bounded degree (roughly <20, but many entries are smaller) and integer coefficients, and I'd like to be able to ...
3
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2answers
577 views

Unimodular Matrix calculation

I know for a given matrix $M$, there exists a matrix $U$ over the integers with determinant $+1$ or $-1$ such that $UM=E$. I know $E$, but $M$ is not a square matrix. Is there any easy way to get $...
3
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1answer
2k views

Computing the Cholesky decomposition based of the QR decomposition

Let A be a n×n positive-definite Hermitian matrix. I already have the QR decomposition of A. Is there an efficient way to utilize this knowledge to speed up the Cholesky decomposition of A?
3
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1answer
3k views

How to change the dimensions of an Eigen Matrix in a loop?

I have a while loop, in which I use a Matrix A, vectors B and x with varying dimensions: <...
3
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1answer
294 views

LU Decomposition with memory-mapped matrices

I have a ~4.12 Tb structured relatively-sparse matrix dataset (about 8% of the matrix entries are non-zero) that i want to apply an LU decomposition, however, given the size of it, loading it in ...
3
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2answers
5k views

how can a 2-d fft be constructed to an equivalent matrix?

When I use the cvx matlab toolbox, I met a puzzled problem. The function of fft (or dct, wavelet, etc.) cannot be recognized by the type of 'cvx'. For the 1-d fft, it can be constructed to an ...