Tagged Questions
0
votes
0answers
20 views
Prove that if $A$ is symmetric and has a LU-decomposition then $A=LDU' \Rightarrow U'=L^T$, where $L^T$
Suppose the matriz $A$ has a LU-decomposition, in other words, suppose there exists matrices $L$ and $U$ such that $A=LU$ where $L$ is lower triangular and $U$ is upper triangular.
We can to prove ...
2
votes
1answer
92 views
Ways of computing $A^\infty$
As a follow-up on this question, I would like to ask which one is the better way of computing $A^\infty = \lim_{n \rightarrow \infty} A^n$.
Repeatedly square, computing $A^2, A^4, A^8$ and so on.
Do ...
2
votes
0answers
40 views
Proving invertibility of matrices using banachs lemma
I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
0
votes
1answer
65 views
matrix with distinct bounded eigen values is bounded?
I am looking from the numerical methods perspective. I have a mapping $G$ that maps points in the numerical iteration to the new level. I would like to show its stability. For that I need to show that ...
0
votes
1answer
75 views
Solve: This System of equations for $X$ (does a real solution, exist?)
How can I solve $AX + diag(X)[I-c]=0$ for $X$?
All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...
0
votes
0answers
73 views
Real representations of Lie algebra $\mathfrak{so}(3)$
How does one construct a real-valued, $n$-dimensional representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
1
vote
1answer
37 views
A basic question about convergence of matrix
I am confused with this very basic question.
We know that for a square matrix A the following two properties are equivalent to A being a convergent matrix:
1: $lim_k\rightarrow \infty \|A^{k}\| = ...
1
vote
2answers
70 views
Finding all $\alpha$ such that a matrix is positive definite
I have
$A = $
$
\left[\begin{array}{rrr}
2 & \alpha & -1 \\
\alpha & 2 & 1 \\
-1 & 1 & 4
\end{array}\right]
$
and I want to find all $\alpha$ such that $A$ ...
3
votes
2answers
124 views
Solution of a Sylvester equation?
I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
0
votes
1answer
35 views
what we can say about block inverse besides schur complement
Suppose I have a matrix $M$, which has a block structure $%
\begin{bmatrix}
A & B \\
B^{T} & C%
\end{bmatrix}%
$, where A has the inverse. How can I better numerically calculate $A^{-1}B$ ? ...
2
votes
3answers
56 views
Solving for positive semidefiniteness
Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
2
votes
0answers
32 views
maximal m-elements of the matrix inversion
Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
2
votes
0answers
143 views
Optimization problem about large matrices
I'd like to solve the following optimization problem:
Find non-negative scalar $a$, $b$, $c$ to minimize
$\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$
where ...
3
votes
3answers
51 views
Raising a matrix to a large power when the values are fractions (precision problem)
I have a matrix $M$ where various elements may be in the form of $x/y$. If I use the decimal form of that number, I lose precision if I raise $M$ to a large power.
My question: is it possible to do ...
4
votes
2answers
104 views
Math hack for solving system of equations
Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
0
votes
0answers
62 views
Is my rectangular matrix dissipative?
I'd like to find out if my rectangular matrix is a dissipative operator. Let me explain how the matrix is formed.
I'm given a set of parameter values $\lambda_i, i=1,2,...,N_d$ with $0\leq \lambda_i ...
2
votes
1answer
135 views
Jacobi's Method for a matrix
Say we have a $2\times 2$ matrix $A$:
$$A=\begin{pmatrix} 1&2 \\ 3&1\end{pmatrix}$$
What is the spectral radius of $A$?
So I get the eigenvalues of $A$, and the maximum eigenvalue ...
2
votes
0answers
67 views
all eigenvalues of a large sparse symmetric matrix
my question is similar to
how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors
however i wish to be more concrete and ask if one can, on a standard PC
(e.g. a ...
2
votes
3answers
29 views
Matrix decomposition again
If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors:
what is proper term for such kind of decomposition?
how to tell if such kind of decomposition exists for given matrix?
...
0
votes
0answers
73 views
Reduce condition number of a matrix
I have a vector with observations $t = [t_1, t_2, ..., t_n]$, for the vector I can construct a matrix $X = \left( \begin{array}{cccc}
1 & t_1 & ... & t_1^p \\
1 & t_2 & ... & ...
4
votes
2answers
92 views
Bounding $x^TAx$ when A is not a symmetric matrix
If $A$ is a real and symmetric $n\times n$ matrix, then we know $\lambda_{min}||x||^2\le x^TAx \le \lambda_{max}||x||^2\ \forall x\in \mathbb{R}^n$ where $\lambda_{max}$ and $\lambda_{min}$ are the ...
0
votes
0answers
53 views
Two variable PCA using gradients
Let $x$ and $y$ be two random variables. Using principal component analysis (PCA), I can find a linear projection making the two variables uncorrelated. PCA solves this problem through an eigenvalue ...
3
votes
0answers
43 views
finding the largest $p$ components of $x$
Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations:
...
0
votes
2answers
54 views
Finding eigenvalues of sparse integer matrix
I need to find eigenvalues of a sparse matrix with integer coefficients. I understand in general this is not done by explicitly computing the characteristic polynomial due to numerical instability, ...
1
vote
1answer
48 views
Help regarding a weird Matrix
Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
1
vote
1answer
163 views
Using permutation matrix to get LU-Factorization with $A=UL$
Let $Q$ be the $n$x$n$ permutation matrix
$$Q=
\begin{bmatrix}
0&0&...&0&1\\
0&0&...&1&0\\
.& \\
.&\\
.&\\
0&0&...&0&0\\
...
0
votes
0answers
54 views
How does one more Efficiently Numerically Solve Multidimensional Problems using Spectral Methods?
As per the title, would you please tell me how to more efficiently solve multidimensional partial differential equations? Other then just tediously writing out the matrix elements manually. If you ...
1
vote
2answers
322 views
Moore-Penrose pseudo inverse algorithm implementation in Matlab
I am searching for a Matlab implementation of the Moore-Penrose algorithm (convertable to C++) computing pseudo-inverse matrix.
I tried several algorithms, "Fast Computation of Moore-Penrose Inverse ...
3
votes
0answers
65 views
Gently push my non-Positive Definite matrix back into the set of Positive Definite matrices
I have a matrix $\eta$ that should be Positive Definite but it is not. Is there a numerical method to gently push my non-Positive Definite matrix back into the set of Positive Definite matrices?
...
-1
votes
1answer
58 views
variant eigenvector problem
I have the following problems when solving a linear equation.
Let $A=(a_{i,j})_{n \times n}$ be a non-negative matrix with $a_{i,j} \in (0,1)$, and let $0<r<1$ be a scalar. Now we define a ...
2
votes
0answers
60 views
A basic question about randomly generated matrix
I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been ...
1
vote
2answers
204 views
Condition number matrix
For the identity matrix $I$, the condition number of the matrix always equals 1.
My question is: are there any other matrices out there that have a condition number equal to 1, but are not the ...
1
vote
1answer
529 views
Existence of non-trivial solution of Sylvester equation.
I'm trying to solve a special case of Sylvester equation
in my case it looks like
$$A*X=X*B$$
so it can be written in form
$$A*X+X*(-B)=C$$ where C consist of all 0 items.
I tried to solve it in ...
1
vote
0answers
101 views
Large number of Linear equation solving with diagonally dominant matrix
For a certain problem I am modelling, I have an MCMC sampler at the moment. It draws samples from the ($n-1$)-dimensional simplex (in this case, from a Dirichlet distribution) and evaluates the ...
5
votes
3answers
237 views
Projection matrices
I have found these two apparently contradicting remarks about projection matrices:
1) A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection ...
0
votes
0answers
257 views
What is the BEST relaxation factor(ω) for Successive over-relaxation method?
I'm working on a project, implementing Successive over-relaxation (SOR) method (http://en.wikipedia.org/wiki/Successive_over-relaxation) using Python. SOR can only apply if given matrix is,
...
1
vote
1answer
120 views
How do I combine two matrix equations into one?
I have a discretely sampled 2D function:
S =
1 2 3 4
1 2 3 4
1 2 3 4
I want to find finite difference ...
2
votes
0answers
80 views
Solution for this matrix equations (closed form or approximate solution)
Given a system of equations, I'm curious whether I can find the closed form solution for $P$,
Here, $G$,$H$ are known $N \times N$ matrix, $\lambda$ is a known scalar;
$s$,$t$ are two $N \times 1$ ...
0
votes
0answers
106 views
a minimum and maximum value problem
First and foremost, I greatly appreciated the prior attempts made by the excellent mathematicians Robert Israel, and mixedmath on the related problem. Now I have the following problems of the ...
1
vote
4answers
460 views
How to solve $Ax = \lambda x + b$ efficiently?
Let $A$ be a real symmetric invertible matrix and $b$ a real non-zero vector. Consider the problem of finding a real number non-zero $\lambda$ and a real valued vector $x$ such that $$Ax=\lambda x ...
2
votes
0answers
117 views
Matrix Exponential with time parameter
Could someone please expand on
Method 9. Lagrange interpolation (page 17) at
http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf
because the summation runs from 0 to (n-1) but the eigenvalues ...
0
votes
3answers
276 views
program for eigenvalue calculation
I have a n x n matrix. I would like to
(a) take successively higher powers of the matrix and then multiply by projection vectors until the resulting vectors differ by only a scalar factor.
(b) ...
2
votes
2answers
78 views
Have spd $(A^TA)$ and $(B^TB)$, need $A^TB$.
Given two symmetric positive definite matrices $(A^TA)$ and $(B^TB)$ I need to compute $A^TB$.
$A$ and $B$ are not given directly.
$(A^TA)$ and $(B^TB)$ have the same dimensions. $A$ and $B$ are ...
0
votes
1answer
334 views
Moore-Penrose pseudoinverse for square matrix
help me please to define Moore-Penrose pseudoinverse for square matrix.
Also, how can i use it in order to solve linear equations ?
Thanks.
2
votes
1answer
212 views
Confusion with “trivial Givens rotations” being used to eliminate values in a vector
I am currently studying the QR algorithm described in Computing the eigenvalues of a
companion matrix and have come to something that has me scratching my head. I'm trying to work this method out on ...
2
votes
2answers
516 views
How do I calculate the derivative of matrix?
I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do
$$ M (T) = M(T_0) + \frac{\partial M}{\partial T} (T-T_0) + \cdots$$
$T$ is a algebraic ...
0
votes
0answers
91 views
A numerical algebra problem. An optimization about matrix
Please give a numerical method to solve the following problem:
Find $X \in \mathbb{R}^{m\times n}$ s.t.
$$\cases{||C-AXB||_F=min,\\||X||_F=min,}$$
Here A,B,C are given, and with dimension $n \times ...
2
votes
2answers
1k views
Can QR decomposition be used for matrix inversion?
Is there any simple algorithm for matrix inversion (that can be implemented using C/C++)?
Can QR decomposition be used for matrix inversion? How?
3
votes
6answers
480 views
Matrix-Square Root
I was wondering about matrix square roots. What is the procedure to evaluate $(X^{T}X)^{-1/2}$? Is it by a spectral decomposition of $(X^{T}X)^{-1}$ as $U\lambda U^{T}$ followed by the square root $S$ ...
0
votes
0answers
251 views
Problem with Newton's Method in solving a System of Equations
I'm trying to use Newton's method to solve the following system of equations, where f and g are functions of x and y. (h,a,f,c,d,b and k are just constants).
$f(y,x)=\left[\begin{array}{c}
y^{1}\\
...
