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

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Why does the Lanczos algorithm make an orthonormal basis for the Krylov subspace?

Starting with a $v_0$= $b_0$=0 and a symmetric positive definite matrix A. Why does the following algorithm forms an orthonormal basis span{$v_1$,$v_2$,...,$v_n$} for $K_n$(A,$v_1$)? for k=1,...,n-1 ...
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59 views

Damped Iteration

For splitting $A = M-P$ a damped iteration with damping factor $\gamma <1$ and scalar $\omega$ is $$x^{k+1} = x^{k} +\gamma M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ $$M =\frac{1}{\omega }I $$ $$P = ...
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25 views

Preconditioning for Jacobi Method without effect

Show that the following scaling doesn't affect the spectral radius of the Jacobi method iteration matrix $T_{J} = -D^{-1}(L+U)$. $\tilde A=D^{-1 /2}AD^{-1 /2}$, where $D = ...
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5 views

how to calculate spectrum of a large collumn stochastic matrix

Okay, I have a collumn stochastic matrix of order $280\times 280$, the entries are given in an url in some webpage in row format. I need to find all the eigen values and eigen vector corresponding to ...
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41 views

what's the difference between Eisenstat trick and an implicit preconditioner?

Assume ${A}$ is Hermitian positive definite and $\hat A$=$D^{-1/2}$$A$$D^{-1/2}$ is to obtain a symmetric variant. and $M$=($L_{A}$+$D$)$D^{-1}$($D$+$U_{A}$) where $D$ is a suitable diagonal matrix ...
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28 views

Square-root of a matrix which arise from truncating a matrix which has a square-root

I have this covariance matrix $A$ which has a symmetric Toeplitz structure. \begin{equation} A = \left[ \begin{array}{cccccccc} c_0 & c_1 & c_2 & \cdots & c_{n-1} & c_{n} \\ c_1 ...
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21 views

Permutation of the posson equation

I have the poisson matrix A, I want to reduce it to the following block matrix: $$ A=\begin{bmatrix} A_{11} & A{12} \\ A_{21} & A_{22} \end{bmatrix} $$, where $A_{11}$ and $A_{22}$ are ...
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14 views

Inverse power method - how to decide which $\alpha$ to be used?

Inverse power method - how to decide which $\alpha$ to be used ? I've learnt who inverse power method run, but I don't know how to choose the $\alpha$ $y=(A-\alpha I)^{-1}u_k $ the $\alpha$ here
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18 views

Prove $\left \| f-L_{k}^{s}f \right \|_{2} = min_{q \epsilon V_{k} }\left \| f-q \right \|_{2} $

Let q be arbitrary and consider the quadratic function of t defined by: $\phi (t)=\left \| f-L_{n}^{s}f+tq \right \|_{2}^{2}$ Note: $L_{k}^{s}f = \sum_{i=1}^{k}(f,p^{i})p^{i}$ for $i = 1,...,k$ ...
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28 views

How to compute the Jacobi matrix (tridiagonal matrix) of a polynomial with a recurrence relationship?

I am looking at Trefethen & Bau Exercise 37.1: I have two normalizations of the Legendre polynomials with corresponding recurrence relations: (1) $P_n(1)=1$ which follows $P_n(x) = \frac{2n-1}{n} ...
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27 views

SVM optimisation problem, finding w

I am finding it difficult to find the value for vector w (weight) for the optimization problem which is: $\min \{ (1/2) * w^T * w : y(i) * w^T * x(i) > 0, \ i = 1,\dots,m\}.$ Can someone ...
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13 views

Let a and $a_1,…,a_m$ be given vectors i n $\mathbb R^n$ .

Show that two statement are equivalent. (a) For all x ≥ 0 , we have $a'x≤ max a'_i􀂂x.$ (b) There exist nonnegative coefficients $b_i$ that sum to 1 and such that $a \le \sum_{i=1}^m b_i a_i $ can ...
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21 views

number of iterations for the generalized conjugate residuals method?

I have the matrix $n \times n $ defined as: $A=\begin{bmatrix} 0 & 1 & 0 & \dots& 0 \\ 0 & 0 & 1 & \dots &0 \\ \dots &\dots ...
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23 views

Generalized conjugate residuals method applied to a block matrix

I have the diagonal block matrix A with $2 \times 2$ k-blocks given by : $D_k=\begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix} $. I have to show that the generalized ...
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21 views

Conjugate gradient method and rietz values

I'm working on the conjugate gradient method. I have the matrix A, defined as A= diag(v) where $v=[ones(1,10), 11:1000]$. I have to solve the system $Ax=b$ ,b=ones(1000,1) with the conjugate ...
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39 views

Find the inverse of $A+uB+vC+uvD+u^2E+v^2F$ where $A,B,C,D,E,F$ are symmetric.

Given scalars $u,v$ s.t. $0<u,v<1$, we seek the properties of the matrix defined by $$P=A+uB+vC+uvD+u^2E+v^2F$$ A is symmetric and positive definite. $B,C,D,E,F$ are symmetric, but might not ...
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10 views

How to find How to find ΔU if LU factorization of tridiagonal Matrix A is used to sovle system Ax=b?

How to find ΔU if LU factorization of tridiagonal Matrix A is used to solve system Ax=b? By using forward and back substitution to show that x̂ satisfies ...
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98 views

How to decide if a system is ill conditioned when the matrix condition number is very different for different norms?

A linear system Ax=b is said to be ill-conditioned if the condition number (A)of the coefficient matrix A is far from 1. Consider the system $$\begin{align}x_1 = &b_1 \\ x_1+x_2 = &b_2 \\ ...
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25 views

How to find Common (invariant) Subspace between more than two Hankel Matrices?

Note: I am not a mathematician but a control engineer. A general nonlinear $n_{a}^{th}$ order discrete-time state-space model is described by the following equations: \begin{align} ...
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20 views

Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique.

Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique. Here is what I have done so far: By the spectrum theorem, suppose ...
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28 views

LU Decomposition simplification on a tridiagonal matrix

If I have a tridiagonal matrix that looks like Tn = diag[1, 3, 1], I can do LU Decomposition of it using n - 1 multiplications (by omitting multiplications with 1) but not n - 1 divisions, right? In ...
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22 views

A problem about the condition number

Given that $I\in R^{n\times n}$ is identity matrix and $||I||=1$.Assumed that Matrix $A\in R^{n\times n}$ is nonsingular,with $\delta A$ satisfying $||A^{-1}||||\delta A||<1$. Then $A+\delta A$ is ...
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Question related to matrix transformation in sequence spaces.

Let $M=[K_{i,j}]$ be an infinite matrix, where $K_{ij}=1/i \text{ if } 1\leq i \leq j \text{ and } K_{ij}=0 \text{ if } i>j\geq 1$. Then $M$ defines a map $\ell^p \to \ell^r$ iff $p=1 \text{ and } ...
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14 views

convergnece of QR-method

I'm studying the QR-algorithm. In particular I have this algorithm: For every $k=0..n $ select a shift $\sigma_k$ factorize $A_k-\sigma_k I =Q_kR_k$ multiply $A_{k+1}=R_k Q_k+\sigma_kI$ muliply ...
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45 views

Compression of a matrix A by V

I can't understand and even can't find any text on Compression of a matrix A by V. meaning if $B=V^*AV$ then B is called the compression of A. What does it mean???
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41 views

Solving for intersection of line and a vector function.

How would one approach a problem of finding intersection points between a line $\vec l = \vec S + d \vec t$ and vector of the form $$\vec v = \begin{pmatrix} x \\ y \\f(x, y) \end{pmatrix}$$ I am ...
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37 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 ...
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71 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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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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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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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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47 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 ...
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33 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 ...
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29 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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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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22 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 ...
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37 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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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 ...
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75 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 ...
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26 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 ...
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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 ...
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57 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. ...
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31 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 $A^{-1}$ ...
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58 views

Approximation of Mahalanobis distance

If $A$ is a symmetric positive definite $n\times n$ matrix then the square Mahalanobis norm of a vector $v\in \mathbb{R}^n$ is given by $$\lVert v \rVert_A^2=v^t A^{-1} v.$$ Now I have a situation ...
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79 views

Proof that Householder Triangularization for QR is backward stable

How do you prove that QR factorization via Householder Triangularization is backward stable? Theorem 16.1 (From Trefethen and Bau): Let the $QR$ factorization of a matrix $A$ be computed by ...
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24 views

Is my QZ-Step as postulated in the paper?

I am working with the paper An algorithm for generalized matrix eigenvalue problems from C.B. Moler and G. W. Stewart (paper available here: ...
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24 views

QZ-Algorithm: how to simultaneously generate the upper triangular form

I need to calculate the Eigenvalues of a generalized Eigenvalue Problem. To achieve this, i wanted to use the QZ-Algorithm, which avoids calculating the inverse of a Matrix. This quite useful, because ...
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75 views

Givens-Rotation from the right

i need to get a Givens-Rotation, which zeros a matrix entry when multiplied from the right side. I did already look at this topic givens rotation from right side but i could not really understand the ...
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14 views

Conversion of cylindrical harmonic field into space-harmonic field for plane waves

It is well known that a plane wave can be represented by an infinite sum of cylindrical wave function of the form $\varphi^i(\rho,\phi)=e^{\left(-j\beta \rho ...
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46 views

PCA when SVD is a skinny SVD

A = m * n matrix. When $m \ge n$, it is easy to see that the V matrix in the full SVD ($A = U*S*V^T$, where U and V are both orthonormal square matrix) and V in a skinny SVD are the same. When $m \lt ...