Questions on the various algorithms used in linear algebra computations (matrix computations).

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29 views

$Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$ L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}. $$ This operator is not ...
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0answers
15 views

Row degeneracy in systems of linear equations

I am trying to understand the concept of row degeneracy in a system of linear equations, but having trouble understanding this problem. \begin{align} x+2y+z &= 2 \tag{1} \\ 2x+y+3z &=5 ...
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0answers
41 views

Solving the linear system $XL + L^TX = M$ efficiently

I'm wondering of an efficient way to solve the following system for the symmetric matrix $X$, given a positive semi-definite matrix $S$ and any matrix $M$: $$ LL^T = S $$ $$ XL + L^TX = M $$ $$ (XL) + ...
1
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1answer
50 views

Using linear algebra (e.g. matrix) methods to solve a system of linear inequalities

Say we have the equation $Ax>b$, where $A$ is an M-by-N matrix, $b$ is a known vector of length N, x is an unknown vector of length N, and the inequality sign means that each element of $Ax$ is ...
1
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2answers
60 views

Positive linear combinations of intervals

Given two intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1=1-a_0<1$ and a third interval $I=[-a,a]$ where $0<a<\frac{1}2$, when is there an $\alpha,\beta\in\Bbb R$ such that ...
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0answers
13 views

Enforcing additional constraints in linear equation

In a finite element context, I come up with a sparse "stiffness matrix" $A$ and a corresponding RHS $b$. The goal is now to solve $$Au = b$$ Where $u$ is a coefficient vector of the solution. Now I ...
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0answers
35 views

Solving System of Linear Equations

These are the two known equations: $$\frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4} = \frac{2x}{L}$$ $$\frac{(I_2+I_4)-(I_1+I_3)}{I_1+I_2+I_3+I_4} = \frac{2y}{L}$$ where I know the values of $(x,y,L)$. ...
0
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1answer
26 views

Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal

also compute $||f-p_2||_2$. Write $p_2$ with respect to $\{P_0,P_1,P_2\}$ and $\{1,x,x^2\}$ I know its helpful to show what I have so far but I really don't know where to start. I'm looking at ...
3
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1answer
108 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A,B \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ A ...
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1answer
55 views

Complexity of Newton iteration problem for a d-dimensional problem

If we assume that we have $f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and we want to use the Newton iteration method to solve $f(x)=0_{\mathbb{R}^{d} }$. Is there any theorem regarding the ...
1
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1answer
23 views

Polar decomposition varient

I have a factorisation to do, and I think that a varient of Polar decomposition will give me what I need, although I'm not sure of the exact form. I have \begin{equation*} \mathbf{y} = ...
0
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0answers
44 views

When to use iterative methods for solving systems of linear equation

Iterative methods such as Jacobi, Gauss-Seidel method and successive over relaxation have a very limited field of use - for diagonally dominant matrices. So how they could be used on practice? What is ...
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0answers
37 views

Implicit Solution of Linear Algebraic Equations with Discontinuities

I am trying to get a reliable algorithm for solving a set of linear algebraic equations involving implicit singularities/discontinuous function. The model equation is: $$ {\bf s}_{n+1} = {\bf s}_n ...
1
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0answers
31 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
0
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1answer
21 views

Linear functional and Hessian

Consider the vector space $\mathbb{R}^n$ provided with the usual inner product $<.,.>$. Let $A\in \mathbb{M}_n(\mathbb{R})$ a invertible matrix, $b\in\mathbb{R}^n$ and $J:\mathbb{R}^n\rightarrow ...
0
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1answer
24 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
2
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0answers
37 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
0
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1answer
17 views

Show if it is lipschitz continuous?

I can't use the mean value theorem to prove this. The problem that I am given is $$ f(x) = (\sqrt{17\pi} )x^2 $$ on the interval $=-10 \le x \le 4$ I know that I have to show that $\lvert ...
0
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1answer
35 views

System of linear equations: and a small perturbation

If $Ax=b$ and $Ax'=b'$ where $x'$ and $b'$ are $x$ and $b$ with a small perturbation, the following inequality will always hold: $ (\left\lVert x-x' \right\rVert/\left / \lVert x \right\rVert) \le ...
0
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1answer
19 views

Find a diagonal matrix D such that the gershgorin disks of the matrix $B=D^{-1}AD$ do not include the origin

I am given that $$ A= \begin{bmatrix} 3 & 4 \\ -5 & 9 \end{bmatrix} $$ Find a diagonal matrix D such that the gerschgorin disks of the matrix $B=D^{-1}AD$ ...
1
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1answer
43 views

Factorization algorithm to solve this system?

What is the best factorization algorithm to solve this system? (Best is intended as more stable) $$ AA^Tx = b $$ x, b vectors
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3answers
29 views

Show that UV is a unitary matrix?

Suppose $U$ and $V$ are unitary matrices of the same size. Show that $UV$ is a unitary matrix. I looked up the definition for unitary matrices in my notes. It says that A matrix is unitary if $UU^*= ...
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0answers
26 views

Smallest square problem, $A^*A$ singular?

In our numerics class, we have to solve the smallest square problem $Ax = b$ with $$A = \left( \begin{matrix} 1 & 3 &-4\\ 3 & 9 & -2\\ 4 & 12 & -6\\ 2 & 6 & 2 ...
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0answers
14 views

Linear Inverse Problem with symmetry constraint

I'm not entirely sure if this is even a solvable problem: $\mathbf{A} = \mathbf{B} \mathbf{C}$ Knowns: $\mathbf{A} \in \Bbb{R}_{n\times m}^{+}$, $\mathbf{B} \in \Bbb{R}_{n\times m}^{+}$ An ...
0
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1answer
53 views

Does $A^{-1}$ exist?

Suppose A is similar to the matrix B given below. $$ B= \begin{bmatrix} 7 & 0 & 0 \\ a_{21} & 4 & 0 \\ a_{31} & a_{32} & -0.5 \\ \end{bmatrix} ...
0
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4answers
36 views

Show that A is diagonalizable?

Show that A is diagonalizable? That is, show that A is similar to a diagonal matrix, D, by finding a matrix P such that D= $P^{-1}AP$. Show all your work. I already found the eigenvalues and ...
0
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0answers
56 views

Determining Nullspace Basis so that only one column is deleted or added as a row is added or deleted, with remaining columns of basis staying 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, ...
1
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1answer
35 views

What is $a_{22}$?

I remember doing this problem in linear algebra where you had to solve for k given the determinant and the rest of the values in the matrix. This problem is a little more complicated. Two of the ...
1
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3answers
52 views

What is the minimum and maximum number of eigenvectors?

I am given the eigenvalues of a square, 8x8, matrix. They are all non-zero. I have determined that the matrix is diagonalizable and has an inverse. In one part of the problem, I am asked to find the ...
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0answers
28 views

the SVD (singular value decomposition) of an augmented matrix

Suppose we have a $4\times 3$ dimensional matrix $A$. Denote the SVD of $A$ by $USV^T$, where $U\in R^{4\times 3}, S\in R^{3\times 3}, V\in R^{3\times 3}$. Then, we construct a new matrix $B=[A;0]\in ...
3
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1answer
36 views

Definition of Distinct eigenvalue clarification?

I'm solving a problem where I am given the eigenvalues of a matrix $A$ and need to solve for the determinant of $A$. I know that if my matrix is diagonalizable I can find the determinant of $A$ by ...
0
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0answers
24 views

Upperbound for a linear algebraic ratio?

Consider ($n\times 1$)-column vector $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ and a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $i \neq ...
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0answers
10 views

Backwards Stability of systems

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that $x^{k+1}$ is $\epsilon$-backward stable approximate of ...
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0answers
28 views

Linearize discretized nonlinear system model

For the following nonlinear system I want to find the linearization after a discretization: $$ \begin{pmatrix} \dot{x_{1}} \\ \dot{x_{1}} \\ \dot{x_{1}} \end{pmatrix} = 1/A \begin{pmatrix} ...
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1answer
54 views

a multiple choice question related to trace of a matrix.

let P and Q are two invertible matrices . and PQ= -QP . then which of the following is true a) trace(P)=trace(Q)=0 c)trace(P) is not equal to trace(Q) c) none of the above. i can show that ...
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1answer
25 views

Any time saving/ short methods to solve this problem?

Three persons A,B,C whose salaries together amount to $144000. Each spend 80,85 & 75 percent of their salaries respectively . If their savings are in the ratio 8:9:20, then C's salary is? ...
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0answers
16 views

sparse, complex, unymetric test-matrix

Can anybody recommend me a sparse, complex, unsymmetric test-matrix (maybe from MartixMarket) which is solvable with a transpose-free QMR without preconditioning in under 1000 iterations?
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0answers
20 views

How many kind of basis function to approximate an arbitrary function

I am finding a list algorithm to approximate an arbitrary function. Such as Bernstein, he said that a linear combination of Bernstein basis polynomials $$B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} ...
1
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2answers
41 views

Problem implementing a QR factorization

I'm trying to write a Fortran subroutine to compute a QR factorization using the Householder method. To test my routine, I compute the factorization of the following matrix: $$ A = \begin{pmatrix} ...
2
votes
0answers
16 views

LU factorization accuracy

I'm doing some experiments with LU factorization (without pivoting). Basically I have a 2x2 matrix and a $b$ vector and I try to solve Ax=b. $A$ looks like: \begin{pmatrix}a&1\\1&1\\ ...
1
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0answers
13 views

Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...
0
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0answers
19 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
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1answer
35 views

Linear Algebra - minimal polynomial, polynomial

the minimal polynomial of $A$ is $(x−1)(x+1)$. Let $f(x)=4x^{2008} − 8x^{597} + 10x + 6$ show $f(A) = \alpha I + \beta A$ $\alpha=?\ \beta=?$ So I worked on a bit, and I got this far $A = ...
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0answers
21 views

How to implement QR method for bidiagonal matrices?

My goal is to take the singular value decomposition of a (not necessarily square) matrix. I have a method to do bidiagonalization of a matrix, and I can chop the bottom rows of zeros. In order to find ...
1
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0answers
22 views

SVD of a block partitioned matrix

Given a block partitioned matrix $\boldsymbol{A}$ $$ \boldsymbol{A} = \begin{bmatrix} \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \cdots \\ \boldsymbol{A}_{2,1} & ...
0
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0answers
36 views

Help understanding Jacobi SVD

I found this link, and I want to complete this implementation of the Jacobi SVD method, but it isn't clear to me how to implement alpha, beta, and gamma. I think it's very clear that ...
2
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2answers
20 views

convergence of iterative methods for linear system

Here is a theorem about convergence of iterative methods for linear system in Burden and Faires' book "Numerical Analysis" For any $x_0 \in \mathbb{R}^n$, the sequence defined by $x^k = Tx_{k-1} + c$ ...
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0answers
6 views

Find rank-vital rows (coloops) of a matrix

Let $A$ be a $m\times n, m\geq n$ matrix over a finite field. Coloop is any row of $A$, such that the rank of $A$ is decreased when that row is removed. What is an efficient algorithm to find all the ...
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2answers
35 views

What is the intuition behind matrix splitting methods (Jacobi, Gauss-Seidel)?

Descent Methods, like Gradient and Conjugate Gradient ones, have a nice geometric interpretation and I really love them. What about Jacobi, Gauss-Seidel or other matrix splitting methods? I can't see ...
0
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1answer
32 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...