# 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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### Symbolic software packages for Matrix expressions?

We know that $\mathbf A$ is symmetric and positive-definite. We know that $\mathbf B$ is orthogonal: Question: is $\mathbf B \cdot\mathbf A \cdot\mathbf B^\top$ symmetric and positive-definite? ...
2answers
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### why is A*v+B*v faster than (A+B)*v?

$A$ and $B$ are $n \times n$ matrices and $v$ is a vector with $n$ elements. $Av$ has $\approx 2n^2$ flops and $A+B$ has $n^2$ flops. Following this logic, $(A+B)v$ should be faster than $Av+Bv$. Yet,...
4answers
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### Dealing with the inverse of a positive definite symmetric (covariance) matrix?

In statistics and its various applications, we often calculate the covariance matrix, which is positive definite (in the cases considered) and symmetric, for various uses. Sometimes, we need the ...
11answers
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### Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
5answers
7k views

### What is the best way to determine the number of non zeros in sparse matrix multiplication?

I was wondering whether there is a fast and efficient method to find the number of non zeros in advance for sparse matrix multiplication operation assuming both matrices are in CSC or CSR format. I ...
2answers
2k views

### Practical example of why it is not good to invert a matrix

I am aware about that inverting a matrix to solve a linear system is not a good idea, since it is not as accurate and as efficient as directly solving the system or using LU, Cholesky or QR ...
3answers
8k views

### Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
4answers
7k 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 ...
4answers
3k views

### Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
3answers
2k views

### Why isn't my Matrix-Vector Multiplication Scaling?

Sorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices. ...
2answers
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### Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
2answers
2k views

### Does the “cofactor technique” for inverting a matrix have any practical significance?

The title is the question. This technique involves using the "matrix of cofactors", or "adjugate matrix", and gives explicit formulae for the components of the inverse of a square matrix. It is not ...
4answers
388 views

### Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...
3answers
4k views

### Single versus double floating-point precision

Single precision floating point numbers take up half the memory and on modern machines (even on GPUs it seems) operations can be done with them at almost twice the speed compared to double precision. ...
5answers
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### Calculation of the sparsity structure for finite element matrices

Question: What methods are available to accurately and efficiently calculate the sparsity structure of a finite element matrix? Info: I'm working on a Poisson Pressure Equation solver, using Galerkin'...
2answers
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### Compute all eigenvalues of a very big and very sparse adjacency matrix

I have two graphs with nearly n~100000 nodes each. In both graphs, each node is connected to exactly 3 other nodes so the adjacency matrix is symmetric and very sparse. The hard part is I need all ...
3answers
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### 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 ...
5answers
2k views

### Repeatedly solving $\mathbf{A} \mathbf{x} = \mathbf{b}$ with same $\mathbf{A}$, different $\mathbf{b}$

I am using MATLAB to solve a problem that involves solving $\mathbf{A} \mathbf{x}=\mathbf{b}$ at every timestep, where $\mathbf{b}$ changes with time. Right now, I am accomplishing this using MATLAB'...
2answers
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### In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
3answers
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### What's the current state of the art regarding algorithms for the singular value decomposition?

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...
1answer
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1answer
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### Solving system of linear equations with cyclic tridiagonal matrix

I have this problem in my textbook: Suggest efficient algorithm for solving system of linear equations with cyclic three-diagonal matrix, that is of the form: \begin{bmatrix} a_1&b_1&0&...
1answer
484 views

### Algorithm to calculate the exponential of an Hessenberg matrix

I am interested in computing the solution of a lage system of ODEs using a krylov method as in . Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
0answers
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### Wanted: sequences of linear systems for recycling Krylov solver analysis

In the solution of sequences of linear systems $$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$ with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
0answers
437 views

### Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy

I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states: $$(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}$$ I wish to perform small ...
5answers
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1answer
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### How to get sparse complex matrices from my code to PETSc efficiently

What is the most efficient way to get a complex sparse matrix from my Fortran code to PETSc? I understand that this is problem dependent, so I tried to give as many relevant details as possible below. ...
1answer
180 views

### Sudden drops in matrix multiplication performance

I've been reading about implementing dense matrix multiplication when the matrix doesn't fit in cache. One of the graphs I've seen (slide 9 from these slides) shows sudden drops in performance using ...