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

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1answer
21 views

Sum of orthogonal matrices

Consider the subspace $\mathbb{R}^m$ with usual inner product.Let $S_1$ and $S_2$ subspaces of $R^m$, $P_1\in M_m(\mathbb{R})$ the orthogonal projection matrix on $S_1$ and $P_2\in M_m(\mathbb{R})$ ...
3
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0answers
16 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
-2
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1answer
22 views

Orthogonality and inner product

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
1
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1answer
18 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
1
vote
1answer
21 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
2
votes
1answer
42 views

Find desired root (based on certain constraints) while using Newton-Raphson method

My function is a vector of dimension 6, and on using Newton Raphson method, the solution usually converges to the nearest root. However, I know that my function has multiple roots (it's an Inverse ...
1
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0answers
30 views

Really confused about LU decomposition and Doolittle algorithm

I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ...
1
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0answers
23 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
7
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1answer
60 views

Singularity of a positive linear combination of rank one matrices

Given a set of rank one matrices $A_1,..,A_n$, we need to find out if there exists $x \in \mathbb R^n$ with $x\gg 0$ (i.e, positive) such that $$ \sum_{i=1}^n x_i A_i = x_1 A_1 + .... + x_n A_n $$ ...
0
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0answers
15 views

QR method for Hessenberg matrices

In trying to implement the method, my approach is to use a reduction to Hessenberg form, and then to iterate using a QR method of Givens rotations. However, I am having trouble successfully ...
1
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1answer
37 views

Convergence of the LR algorithm for $2\times 2$ SPD matrices

I've been asked to prove that the following iterations converge to the eigenvalues of SPD $A_0 \in \mathbb{R}^{n \times n}$ $A_0 = \begin{bmatrix}a & b\\ b & c \end{bmatrix}$ with $a \geq ...
0
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1answer
13 views

efficient computation of Cholesky decomposition during tridiagonal matrix inverse

I have a symmetric, block tridiagonal matrix $A$. I am interested in computing the Cholesky decomposition of $A^{-1}$ (that is, I want to compute $R$, where $A^{-1}=RR^T$). I know how to compute the ...
0
votes
1answer
29 views

Relationship between eigenvectors of two matrices

Suppose I have matrix $A \in R^{2n \text{x} 2n} $ given by $X^{-1} diag(W - iY, W + iY) X$ and matrix $B \in C^{n \text{x} n}$ and $B = W + iY$. Let $v$ be an eigenvector of $A$. How can I relate ...
1
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1answer
32 views

Hessenberg reduction

Given $A \in \mathbb{R}^{nxn}$ and $z \in \mathbb{R}^n$, find orthogonal $Q$ such that $Q^TAQ$ is upper Hessenberg and $Q^Tz = \beta e_1$. My attempt so far, Individually I can find the Householder ...
2
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1answer
69 views

double root and newton method, a problem on solved exercise? [on hold]

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
1
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1answer
59 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
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2answers
30 views

solving given linear equation

So before you guys judge me, I honestly am so clueless with this so please bare with my dumb questions. I have been at this equation that I've been going for an hour now, $\frac{(3x-1)}{2} -2 = ...
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0answers
43 views

Positive solutions to $A^T A x \geq 0$ [closed]

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. $A$ is an arbitrary matrix. I was wondering if there is a general solution. EDIT: One special solution is when $A^TA$ is row ...
3
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0answers
56 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
0
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2answers
43 views

How to minimize $w^{T}Aw$?

$A$ is $n\times n$ matrix. Find a $w$ ( $n$-dimensional unit vector) which minimizes this function. By $w^{T}$, I mean $w$-transpose. I understand there would be non-linear optimization techniques ...
0
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2answers
25 views

Decomposition of a diagonal matrix into a product of particular matrices.

Could you tell me please, if it's possible to find a decomposition of a diagonal matrix $ 3 \times 3 $ : $ D ( \lambda_1 , \lambda_2 , \lambda_3 ) = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 ...
0
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0answers
18 views

Generate duplicate element from a matrix by formula $b(i:j)=A(i:j,:) \times A^{-1} \times b$

I have an interesting question about generate duplicate elements from matrix. I assume that I have a matrix A (such as the bellow example $5 \times 5$) and vector $b$ is $5 \times 1$. My goal is make ...
1
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1answer
60 views

Properties shared by similar and unitary similar matrices.

We know that matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A=PBP^{-1}$ and they are unitarily similar if $P$ is unitary ($PP^*=P^*P=I$). I want to know : What ...
2
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1answer
46 views

SVD and transpose of a skinny matrix

Show: If $\mathbf{A}\in\mathbb{R}^{M\times N}$ with $M\geq N$, then there exists a matrix $\mathbf{G}$ with orthonormal rows so that $\mathbf{A}^T=\mathbf{G}\mathbf{A}\mathbf{G}$. I'm pretty lost on ...
0
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1answer
31 views

Deducing a formula for multiplying a tri-diagonal symmetrical matrix with vectors

This is more like a math-programming problem, dealing with memory efficiency, but I thought it would be nice to expose it here. Let $A \in \mathbb{R}^{n \times n}$ be a tri-diagonal symmetrical ...
0
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1answer
53 views

PageRank (power iteration method) convergence rate?

I could not get my head around the idea that the second eigenvalue is the convergence rate. Since the matrix in this application is a Markov matrix (rows/columns sum to one), the largest eigenvalue ...
1
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1answer
23 views

How page rank relates to the power iteration method

I do not see how pageRank relates to the power method. Since for the pageRank we are looking for the steady stable state (vector) for a Markov (transition) matrix and the matrix has already an ...
4
votes
1answer
38 views

Laplacian solvers for inversion of large matrices?

I have a large matrix L of size 400,000 $\times $ 400,000 . I'm using this L matrix in the following way. Lin = L$^{-1}$ C = D - B * Lin * B'; B,D are of appropriate sizes. L matrix is ...
0
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2answers
39 views

Generalized formula for sum of products.

Q:The sum of all possible products of the first n natural numbers taken two by two is? I did not understand the question as it is.What exactly is being asked?I'd really appreciate an answer ...
0
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0answers
16 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
0
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2answers
33 views

norms of Symmetric Positive Definite Submatrices

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
2
votes
2answers
34 views

Cholesky Factorization with submatrices

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
0
votes
1answer
24 views

Principal Submatrices of a Positive Definite Matrix

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
1
vote
1answer
33 views

showing condition number of a matrix is the square root of $A^T A$

For $A \in \mathbb{R}^{m\times n} : m > n, A$ has full rank, I want to show that $k(A^T A) = k(A)^2$, is there a way to do so purely from $k(A)=norm(A) norm(A^\dagger)$? Recall that $A^\dagger = ...
1
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0answers
31 views

Using Cholesky factorization to solve the system AXA=B

I have been given a problem of solving X, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1}$$ Where the matrix A describes the blurring of an image and the matrix B is ...
2
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0answers
32 views

Nearest non-negative solution for $Av=b$

Let $A$ be a $n\times m$ matrix. Let us define the system $$Av=b$$ $$v\geq 0$$ I want to find a solution $v$ of this system that is the closest (euclidean norm) to $v_0$, a given $n$-dimensional ...
0
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2answers
39 views

Better Gaussian Elimination for solving $Ax=b$ [closed]

We know that Gaussian Elimination is very popular method to resolve $Ax=b$. Does anyone know better method than Gaussian Elimination in term of time complexity? Second question,if I assume that A is ...
0
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1answer
52 views

Is $ \frac{ x^T A A x }{ 1+ x^TAx} $ is upperbounded by the biggest eigenvalue of $A$?

I read somewhere that $$ \frac{ x^T A A x }{ 1+ x^TAx} $$ is bounded by the biggest eigenvalue of $A$, where $x \in \mathbb{R}^d$ and $A \in \mathbb{R}^{d \times d}$ and it is PSD. Anyone see why ...
0
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0answers
40 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
3
votes
1answer
41 views

What is the upper bound on the error of the solution of a noise perturbed linear system $Mx=b$?

Let $x$ be solution to the following linear system: $$ Mx = b$$ and let $ \tilde{x}$ be the solution to the above linear system with some additive noise: $$ M \tilde{x}= \tilde{b}$$ where ...
0
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1answer
38 views

Very High degree Polynomial Roots: How to Cope with Large Values?

I hope I'm not duplicating! I'm wondering how it is possible to find all roots of a polynomial of very high degree (100,1000,1000000, ...) numerically. In all numerical methods, the polynomial is ...
2
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0answers
63 views

Express Lagrange polynomial in term of Cauchy matrix

Given 2n distinct real numers $s_1,s_2, \dots, s_n$ and $t_1, t_2, \dots,t_n$ define the $n \times n$ Cauchy matrix $C = C(t,s)$ by $C_{ij} = \frac{1}{t_i - s_j}$. Express the Lagrange interpolation ...
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0answers
28 views

Normal form calculation

I am working on a problem involves 4 dimensional dynamical system. Is there any ready package (for maple ,matlab...) which calculate the normal form of nonlinear continuous dynamical systems? The ...
2
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1answer
39 views

Solving $Ax_2 = \lambda x_1$ and $A^Tx_1 = \lambda x_2$ using SVD

Please using only SVD, I have solved the problem using other methods Solving $Ax_2 = \lambda x_1$ and $A^Tx_1 = \lambda x_2$ using SVD: I am solving this to find $\lambda$ and $x_1,x_2$ To find ...
0
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2answers
50 views

Prove that I -xx* is singular if and only x*x = 1

(=>) Suppose I - xx* is singular if and only there is a y such that (I−xx*)y=0, i.e. xx* y=y. Now set λ=x* y. Then y=λx, i.e. xx* λx=λx Thus λx(x* x) = λx => x* x = 1 (<=) Suppose x*x = 1 ...
0
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1answer
29 views

Normal Equations error bounds

$A^TAx = A^Tb$ $A^TA\hat{x} = A^Tb + f$ where $\lVert f\rVert \leq cu\lVert A\rVert\lVert b\rVert$ Show that $\frac{\lVert x-\hat{x}\rVert}{\lVert x\rVert} \leq cuK(A)^2\frac{\lVert ...
1
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1answer
31 views

Thomas Algorithm for Tridiagonal System

A professor gave us an assignment to solve a Tridiagonal system using Thomas Algorithm. Here is the exercise: I am lost as to what to do with that $(0.2\pi)^2$ and do I just calculate the ...
3
votes
1answer
47 views

Understanding the QR eigenvalue finding algorithm

I'm trying to code up a matrix library (purely as a learning exercise). This question is about the math I'm trying to understand in order to implement it. I just want to make sure I have a firm grasp ...
0
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0answers
34 views

Can someone explain how to obtain zeroes for L and U for A=LU factorization?

I understand that,In A=LU, for the L = lower triangular matrix, must have zeroes for all elements above the main diagonal and for U = upper triangular matrix, we need to have all elements as zeroes ...
1
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1answer
31 views

Proof of an alternate Matrix Condition Number Representation

I'm currently looking over a section in my textbook on Matrix Condition Numbers and it's given the definition $cond(A) = ||A|| \cdot ||A^{-1}||$ but it's also equated this definition of a condition ...