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

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1answer
64 views

Algorithms for computing matrix logarithm.

On my quest to find the holy grail of mathematics become a little bit better at algebra, I have read up on matrix logarithms and exponentials and how useful they can be in investigating groups and ...
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1answer
38 views

Equaity of two norms of matrix

I have to prove that if A is a $n \times k $ matrix and $A^H$ the hermitian matrix of A, $||A||_2=||A^H||_2$. Where $||A||_2=\sup\{ ||Ax||: ||x||\leq 1\}$ and $|| \cdot ||$ is the euclidean nrmm of ...
2
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1answer
42 views

equality of norms

I have to show that $\| A\|_2=\sqrt{\| A^H\times A\|_2}$. $A$ is a $n\times k$ matrix, $\| \cdot \|_2$ is defined as : $$\| A\|_2=\sup\{ \|Ax\|, \|x\|\leq 1 \}$$ and $\| \cdot \|$ is the ...
0
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2answers
34 views

A proof of the uniqueness of svd?

I understand the geometric intuition, but the proof by induction in Trefethen book confuses me : it seems to me that a 1*1 complex matrix has infinitely many left and right singular vector pairs? ...
1
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0answers
38 views

Computational methods to minimizing the norm of a matrix monomial.

Linear optimization solves the problem $$\min_{\bf x}\{\|{\bf Ax - b}\|_2^2\}$$ Edit: Some clarification Doing the derivation of the optimum, first expand the norm: $$\|{\bf Ax - b}\|_2^2 = ({\bf ...
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0answers
35 views

Rapid calculation of eigenvectors for a submatrix when I know the eigenvectors of the full matrix

I have an $N \times N$ matrix $Z$ that is complex-valued and nonsymmetric in general. I can solve for the eigenvalues and eigenvectors of this matrix numerically. Call the diagonal eigenvalue matrix ...
2
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1answer
62 views

To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a ...
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0answers
66 views

Extracting I-Vectors from a GMM-supervector

The probability model for I-vector in GMM-UBM system is : M = M(ubm) + Ty where, M is GMM supervector (means of all gaussian components concatenated in single vector) M(ubm) is UBM supervector I ...
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1answer
47 views

How to substitute and matrix into a functions?

I have $f(x)=2*x_1 +x_2$ how to find $f(m*x)$ if m is a matrix $m=\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}$
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0answers
10 views

Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its ...
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0answers
48 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
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0answers
23 views

Speeding up Conjugate Gradients iterations for Sparse Matrices?

I've been using Conjugate Gradients to minimize linear systems involving sparse matrices. Although many of my sparse matrices are highly specialized - i.e. for any given row it is easy to know which ...
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0answers
42 views

Finite difference: Radial symmetry boundary condition in tridiagonal system?

I am putting together an axisymmetric finite difference solver for Poisson's equation over a non-"rectangular" boundary in axisymmetric cylindrical coordinates. I was planning on using the dynamic ...
0
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0answers
29 views

Error bounds for solution of system of linear equations when coefficients are uncertain

I have a square system $Ax=b$ and would like to know how much the solution $x$ can change when I change the coefficient matrix $A$. I've stumbled upon the condition number, but this seems to apply ...
4
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1answer
69 views

What mathematics topics pertain more towards applied mathematics?

I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture ...
0
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1answer
44 views

solve nonlinear system of equation numerically

solve the following system of equations numerically $$2x+2y - e^{xy} = 0$$ $$x^3 + y - xy^3 = 1$$ I'm also asked to solve analytically but I'm pretty sure the closed form solution doesn't exist ...
1
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1answer
40 views

Improving the performance of eigs for a large spd Problem

I have two large (think around $100.000\times 100.000$), sparse, real symmetric and positive definite matrices $A$ and $B$ and I want to find the smallest generalized eigenvalue $$Ax = \lambda_{\min} ...
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0answers
597 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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1answer
32 views

Stability of (floating point) computed variance

Homework Question from Accuracy and Stability of Numerical Algorithms, 2nd Edition, by Nicholas J. Higham, page 33: So every time we store an number and do a operation, we introduce an error bounded ...
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0answers
27 views

What are some possible reasons for a large condition number?

For this question, please assume that I am talking about the condition number with respect to the spectral norm. That is, $\kappa_2(A) = \|A\|_2\|A^{-1}\|_2 = \frac{\sigma_{max}(A)}{\sigma_{min}(A)}$. ...
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0answers
12 views

Proof of QR Algoirthm Convergence

I am reading Trefthen and Bau and the amount of implicit proof steps are killing me. Can someone explain how the statement of convergence for the "pure" QR Eigenvalue Algorithm (Theorem 28.1) is ...
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0answers
18 views

how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
0
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0answers
36 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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0answers
12 views

Linear Relationship of two equations

if $0.46a = 120b$ and $2.68a = 60b$ The relationship is linear. what does $0b$ equal in terms of $a$? what does $1b$ equal in terms of $a$? A method to work this out would also be nice, I have ...
1
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1answer
28 views

Gauss Seidel Iteration for a specific matrix

We seek to solve $Au= f$ via iteration, where $$ A = \left ( \begin{array}{cc} I & S \\ -S^T & I \end{array} \right ) $$ Where $S$ is an arbitrary square matrix in $R^n$ and $I$ is the ...
4
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1answer
62 views

LU decomposition for cyclic tridiagonal matrices

It is known that a tridiagonal matrix $$ A = \begin{pmatrix} b_1 & c_1 & 0 & 0 & \dots & 0\\ a_2 & b_2 & c_2 & 0 & \dots & 0\\ 0 & a_3 & b_3 & c_3 ...
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0answers
40 views

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $?

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $? In the other words, is there a way to write $a_i$ in terms of ...
0
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0answers
70 views

Most stable algorithm to solve a system of linear equations?

I am doing some image processing involving solving a system of linear equations. I am getting some errors and bits of the image looks corrupted. I would like to know what is the most stable way to ...
1
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1answer
49 views

Strange behavior with coordinate transformation of square and quadrilateral

I am trying to map coordinates from a quadrilateral to a square. The coordinates are square: $(500,900)(599,900)(599,999)(500,999)$ quad: $(454,945)(558,951)(598,999)(499,999)$ where the $i^{th}$ ...
0
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1answer
31 views

Notation regarding linearization near equilibrium point of dynamical system

Suppose we have $\frac{dx}{dt} = \dot{x} = f(x)$ with equilibrium point $x_e$ such that $f(x_e) = 0$. Then for the linearized approximation of the differential equation near $x_e$ we hope to use the ...
1
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1answer
52 views

How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...
0
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1answer
56 views

Bound on the difference of matrix diagonals

I have two diagonal matrices $\Lambda,\hat{\Lambda}\in\mathbb{R}^{n\times n}$ with non-negative diagonal elements. And I have two matrices $W,\hat{W}\in\mathbb{R}^{m\times n}$, with $m\geq n$, each ...
0
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0answers
25 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
3
votes
2answers
141 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
0
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1answer
177 views

Product rule in discrete derivative in finite difference scheme.

Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define $$ \Delta_+u(i) = \frac{u(i+1)-u(i)}{h} $$ as the ...
1
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2answers
52 views

Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
1
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0answers
26 views

Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
0
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1answer
32 views

Gauß-Newton Example with one variable

$$T=f(t):=2 \alpha + \sqrt{\alpha^2+t^2}$$ To estimate $\alpha$ we got the measured values $T_i$ for $t_i$. Formulate the curve fitting problem and show each step in the Gauss-Newton algorithm. My ...
3
votes
3answers
84 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
1answer
58 views

Choose $\rho$ such that $\rho$-norm minimizes the matrix condition number

I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes. Question Let $A$ a symmetric ...
0
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1answer
44 views

Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
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0answers
38 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
1
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1answer
234 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
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0answers
117 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
0
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1answer
58 views

Can adding one column to a matrix increase its rank by more than one

Knowing the answer to this question would help me answer the following question: $A$ is an $m\times n$ matrix with $m>n$, and let $A=\hat{Q}\hat{R}$ be a reduced QR factorization. Suppose ...
0
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2answers
45 views

Rewrite matrix equation as a quadratic programming problem

Given real-matrix $X_{n\times p}$ how can the problem of minimizing $Tr(X^TA_{n\times n}X)$ under the constraint $Tr(X^TC)=\phi$ be posed as a standard convex quadratic program given by the form: ...
0
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0answers
56 views

Efficient method to compute grand sum of a Vandermonde matrix?

Is there a computationally efficient method to calculate the sum of all elements (grand sum) of a Vandermonde matrix? Each row can be quickly calculated using the formula for a geometric progression. ...
2
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0answers
76 views

Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
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0answers
29 views

Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
2
votes
2answers
81 views

Differential equation for finding closest point on surface.

Inspired by this question I got to think about a more general case. Say I have any discretized surface and want to find closest point from each point outside of surface to the surface. Say that I can ...