All Questions
Tagged with linear-algebra na.numerical-analysis
176 questions
0
votes
0
answers
159
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How to solve a non-local self-consistent equation
I have been struggling lately with solving numerically an equation of the form:
$$ g(x\pm x_{0}) = F[ g(x) ] $$
where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
- 63.9k
1
vote
0
answers
126
views
Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices
I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$...
- 5,380
11
votes
1
answer
3k
views
Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
- 12.9k
2
votes
0
answers
2k
views
How to find a positive solution to an under-determined linear system (if such a solution exists)?
Like the title says, if an under-determined system of linear equations does have at least one positive solution, how to find it efficiently?
Suppose we have an under-determined system:
$$Ax = b$$
...
CommunityBot
- 1
12
votes
2
answers
8k
views
Is there a way to simplify block Cholesky decomposition if you already have decomposed the submatrices along the leading diagonal?
Let's say we have a block matrix $ M =\left( \begin{array}{ccc}
A & B\\
B^{*} & C \end{array} \right)$ where $M$ is positive definite. ($A$ and $C$ are also positive definite.)
There is a ...
- 63.9k
1
vote
0
answers
448
views
Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
- 151
6
votes
2
answers
2k
views
Computation of a Drazin inverse
I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron value)...
5
votes
1
answer
644
views
A conjecture about the submatrix of orthogonal matrix
Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...
- 651
12
votes
5
answers
9k
views
Solving Lyapunov-like equation
The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
- 20.2k
4
votes
1
answer
413
views
Lipschitz property of matrix function only depending on singular values
Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
- 7,514
4
votes
2
answers
3k
views
Methods of solving linear system of equations, how to select the appropriate method
A linear system of equations Ax=b can be solved using various methods, namely, inverse method, Gauss/Gauss-Jordan elimination, LU factorization, EVD (Eigenvalue Decomposition), and SVD (Singular Value ...
- 141
10
votes
2
answers
2k
views
Is there a standard name for (non-square) matrices with orthonormal columns?
One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name ...
- 31.5k
2
votes
1
answer
90
views
Can a Multilayer Perceptron fit any binary function?
Consider a perceptron $F(x) = \phi(x * w - b), \ x \in \mathbb{R}^n,$ (with Heaviside activation function $\phi$) and a dataset consisting of a finite subset $\Omega \subseteq \mathbb{R}^n$ with ...
- 9,501
6
votes
1
answer
218
views
Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?
We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
- 12.6k
0
votes
1
answer
540
views
Computing spectrum of convex combination of SPD matrices given individual spectral decompositions
Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
- 133
7
votes
1
answer
356
views
Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
- 188k
109
votes
19
answers
38k
views
Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
- 52.3k
1
vote
1
answer
102
views
Polynomial Eigenvalue Problem with few non-zero coefficients
Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
- 20.2k
4
votes
1
answer
1k
views
Inverse of matrix with blocks of ones
It seems that there is a nice inverse for matrices that can be written as a diagonal matrix plus a symmetric matrix consisting of scaled blocks of ones.
Consider a real matrix of the form:
$$\begin{...
- 128k
8
votes
2
answers
950
views
Best known bounds on (border) ranks of small matrix multiplication tensors?
The $(m,n,p)$-matrix multiplication tensor is a representation of the bilinear map $T\colon\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\rightarrow\mathbb{R}^{m\times p}$ given by $T(A,B)=AB$. ...
- 31
1
vote
0
answers
125
views
smallest singular value over invertible sub-matrices
Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite.
...
- 303
4
votes
3
answers
283
views
Iterative matrix inversion with $L^\infty$ norm
The usual conjugate gradient type algorithms for iteratively finding the inverse of a matrix applied to a vector, $x = A^{-1} y$, works by minimizing $\|Ax - y\|^2$ where $\| \cdot \|$ is the $L^2$-...
- 28.6k
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
4
votes
2
answers
383
views
Question about preconditioning
I posted the following question on stackexchange but didn't get any replies; I'm hoping perhaps someone can help me here.
I understand that for many iterative methods, convergence rates can be shown ...
- 20.2k
1
vote
0
answers
556
views
Sign correction for SVD in Matlab [closed]
I have a Matlab function that runs a SVD. Unfortunately, the function [U,S,V] = svd(A) has a sign ambiguity which could give misleading results in my application. ...
- 11
1
vote
0
answers
286
views
Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
- 11
10
votes
4
answers
2k
views
How to solve Ax=b incrementally ?
Hi, everyone.
What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
- 25.8k
2
votes
0
answers
193
views
How to find moment condition for generalized method of moments?
Consider a scalar system with $2K$ outputs and $K+2$ unknowns:
$y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables.
To estimate $a_1$ and $...
- 8,529
1
vote
0
answers
174
views
Negative eigenvalue of Toeplitz Hermitian matrix?
I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
- 495
8
votes
2
answers
2k
views
Algorithm for solving systems of linear Diophantine inequalities
So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
CommunityBot
- 1
10
votes
1
answer
449
views
How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...
- 3,377
1
vote
0
answers
90
views
Separable Least squares - is there a notion of conjugate directions?
I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...
- 61
1
vote
1
answer
940
views
Uniqueness and invariance of the LDLT decomposition
A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
- 20.2k
1
vote
0
answers
227
views
Find optimal value for a regularization parameter in generalized eigenvalue problem
Consider the generalized eigenvalue problem :
$ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $
where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices $X$...
- 45
1
vote
1
answer
896
views
sign-flipping inverse
Consider this matrix:
$Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$
Its inverse is entrywise negative (you can check...) and ...
CommunityBot
- 1
12
votes
2
answers
5k
views
Why Householder reflection is better than Givens rotation in dense linear algebra?
It’s obvious that Givens rotation works better with sparse matrices. But I don’t know why Householder reflection is better for dense matrices. Does it require less computations? Or it’s numerically ...
CommunityBot
- 1
7
votes
1
answer
449
views
Can I find the gap between the two least eigenvalues of this special matrix A(t)?
I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal non-...
- 103
0
votes
1
answer
338
views
Is spectral properties a general term for condition number?
I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...
- 20.2k
0
votes
2
answers
1k
views
The condition number of a scaled Vandermonde matrix
Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...
- 2,361
0
votes
1
answer
307
views
Comparing iterative methods for linear systems
For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & -...
- 3,085
4
votes
0
answers
233
views
Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same
I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e.,...
CommunityBot
- 1
7
votes
3
answers
3k
views
Algorithm for the smallest (algebraic) eigenvalues of a symmetric (sparse) matrix
Hi,
I'm looking for a way to get the negative eigenspace of a large (sparse) symmetric matrix. This matrix is basically a discretized version of the operator $-\Delta + V$, $V$ negative, on some ...
- 111
1
vote
0
answers
98
views
LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]
For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
- 111
0
votes
0
answers
83
views
Bits of precision matrix reconstruction
We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
- 13.9k
0
votes
1
answer
98
views
What is exponentially fitted osculating straight line?
While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...
- 188k
2
votes
2
answers
607
views
Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections
I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...
- 3,741
4
votes
1
answer
538
views
Rank 1 Approximation of Elementwise Inverse Matrix
I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...
- 3,741
4
votes
1
answer
396
views
Resolvent of a triangular matrix
Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
- 20.2k
8
votes
0
answers
481
views
Problems where Conjugate gradient works much better than GMRES
I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
- 570
7
votes
1
answer
197
views
Compute only selected components of an eigenvector
I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
- 570