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

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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 ...
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1answer
94 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 ...
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37 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 ...
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1answer
34 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 ...
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44 views

Conditioning of Triangular Matrices:

Let $U \in \mathbb{R}^{N\times N}$ be upper triangular. $U$ is well conditioned if the magnitude of the diagonal elements is sufficiently large compared to that of the corresponding off-diagonal ...
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1answer
39 views

Getting translation and rotation from resultant matrix

I have a matrix which performs a 2D rotation around any given center. Using homogenous coordinates, I have the matrices: $$ T = \begin{pmatrix} 1 & 0 & C_x \\ 0 & 1 & C_y \\ 0 & 0 ...
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1answer
19 views

Coordinate transformation (or conversion) into yards

Following is a soccer field with its dimensions. There is a similar field, but I am capturing coordinates via mouse-movement. So, what (115,75) shows here, is ...
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0answers
99 views

QR fatorization for tridiagonal matrices

Let $$A = \left[\begin{array}{rrrr} \delta_1&\gamma_2 & &0 \\ \gamma_2&\delta_2 &\ddots & \\ &\ddots &\ddots &\gamma_n \\ 0 & &\gamma_n ...
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41 views

prove that $|\lambda(H) - \lambda(B)| \leq \sqrt{||(C^HC)||_2}$

Let A, B be Hermitian square matrices and $$H = \left[\begin{array}{rr}A & C \\ C^H & B\end{array}\right]$$ Show every eigenvalue $\lambda(B)$ of B, there is an eigenvalue $\lambda(H)$ of H ...
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1answer
21 views

a matrix inverse problem

Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to ...
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1answer
22 views

Proof that strictly tri-diagonally dominant matrix has an inverse

We are given the following theorem of which we need only know the result. Theorem Suppose an $n\times n$ matrix $A= (a_{ij})$ is tri-diagonal with $a_{i,i-1}a_{i,i+1} \neq 0$, for each ...
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1answer
44 views

How does one prove the solution of minimum Euclidean Norm to the least squares problem?

If we have some $m \times n$ matrix $A$ with an $m$-vector $b$, how does one prove that the solution $x$ of the minimum Euclidean norm to the least squares problem $Ax \approx b$ is given by $$ x = ...
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52 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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27 views

the rank of QR decomposition

I saw this in a paper, where one has a QR decomposition $C=QR$ ($C\in R^{m\times r}$, $Q\in R^{m\times r}$ is column orthogonal, $R\in R^{r\times r}$, $m>r$). However, under the condition that the ...
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1answer
43 views

Relative error of floating point in inner product

Prove that the floating point arithmetic with machine epsilon $\epsilon$ produces an inner product satisfying: $$\text{fl}(x^Ty) = x^T(y+e)$$ where $$|e_i| \leq 2n\epsilon|y_i|$$ as long as $n\epsilon ...
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1answer
39 views

Question on the spectral radius, regular splitting, and non-singularity/non-negativity

Given $A$ in $R^{nxn}$ and its regular splitting M and N (A = M - N), $M$ is nonsingular and $M^{-1}$ and $N$ are nonnegative. If the spectral radius $p(M^{-1}N)<1$, show $A$ nonsingular and ...
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3answers
69 views

How does one find the reduced Singular Value Decomposition of a row or column vector? [duplicate]

If we treat a column vector $a$ as an $n \times 1$ matrix, or a row vector $a^T$ as a $1 \times n$ matrix, how would one write out the reduced singular value decomposition of $a$?
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4answers
58 views

Is it possible to generate $A$ from a linear system of the form $Ax=b$ given $x$ and $b$?

Take a linear system of the form $Ax=b$. Usually, for obvious reasons, we want to find $x$ given $A$ and $b$. As you all know, the solutions may not be unique, or exist, and many algorithms have been ...
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1answer
92 views

Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
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1answer
56 views

How to find Housholder reflection

For example, let say I have a matrix$$ \left(\begin{array}{rrr} 3 & 3 & 0 \\ 0 & 0 & 0 \\ 4 & 1 & 3 \end{array}\right) $$ and the Householder has the form $H = I -2uu^T$, and ...
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24 views

Iterated Schur complement for block matrices

Suppose you have got a symmetric block matrix $A = \begin{pmatrix} A_{1,1} & \dots & A_{1,n} \\ \vdots & & \vdots \\ A_{n,1} & \dots & A_{n,n} \end{pmatrix}$ Suppose that ...
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1answer
83 views

Singular Value Decomposition using Jacobi Method

First time user of the site, so I apologize if my question isn't worded properly. I'm trying to implement the SVD of a square matrix using Algorithm 6 found on this website: ...
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1answer
43 views

$LDL^T$ decompositon of a symmetric matrix and a matrix determinant expression for the lower triangular entries

Let $n$ be a positive integer, and let $M$ be an integral, symmetric, nonsingular matrix. As $M$ is nonsingular, there exists an $LDL^T$ decomposition such that $D = (d_j)$ is diagonal and ...
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73 views

Prove $\min_{i}|\lambda_i| \leq |r_{jj}| \leq \max_{i}|\lambda_i|$

Let A be a normal $n \times n$ matrix with the eigenvalues $\lambda_1,...,\lambda_n$ |A| = |QR|, $|Q^HQ| = I$, $|R| = [r_{ik}]$ upper triangular matrix. Prove: $$\min_{i}|\lambda_i| \leq |r_{jj}| \leq ...
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43 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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2answers
84 views

Prove that $q_{ki} = \lambda_1[1+ \mathcal{O}((\frac{\lambda_1}{\lambda_2})^k)] \; \text{for all } i \; \text{with} \; (x_1)_i \neq 0$

Let A be a real symmetric $n x n$ matrix having the eigenvalues $\lambda_i$ with $$|\lambda_1|>|\lambda_2| \geq ... \geq |\lambda_n|$$ and the corresponding eigenvectors $x_1...x_n$ with $x_1^Tx_k ...
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1answer
25 views

Reformulate this system of equations

I have the following systems of equations: $$ A\cdot g = \mathbb 0\\ G\cdot \mathbb 1 = w$$ $A$ is a $J\cdot I \times J\cdot I$ matrix. $g$ is a $J\cdot I \times 1$ column vector $\mathbb 0$ and ...
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1answer
66 views

Relationship between the solution to $Ax=b$ and $(A+I)x=b$

I have have a symmetric, tridiagonal, Toeplitz matrix $A$, where $A_{11} = -\frac{1}{2}$ and $A_{21} = 1$, and I need to solve the system $$ (A+I)x=b, $$ numerically where $b$ does not necessarily ...
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28 views

Trying to show convergence of a Forward Euler method based on step size restriction

I have shown that for the given ODE system, that when we apply the forward Euler method to something like \begin{align} \mathbf{y'} &= A\mathbf{y} \\ \mathbf{y}(t_{0}) &= y_{0} \\ t &\in ...
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1answer
120 views

Value of an integral involving the fractional part function

I have difficulties in evaluating the double integral defined in the following. Let $$\left\{ t \right\} = t - \lfloor t \rfloor, $$ $ t> 0$ be the fractional part function, where the ...
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12 views

Powers of matrices via the generalised Lanczos process

At each iterative step of the generalised Lanczos process for the pair of matrices (A,B), we obtain the following factorisation: $$ A Q_k = B Q_{k+1} \widehat{T}_k, $$ where $Q_k^T B Q_k = I_k$ and ...
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1answer
37 views

Technique to calculate rank of this matrix.

1) How can we calculate the rank of this matrix for $n > 3$ ? \begin{matrix} 1^2 & 2^2 & 3^2 & \cdots & n^2 \\ 2^2& 3^2 & 4^2 &\cdots & (n+1)^2 \\ ...
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44 views

Which of the following fixed point iterations will converge?

Which of the following fixed point iterations will converge? Why? Give the rate of convergence. (a) $x_{n+1} = \cos x_n$ (b) $x_{n+1} = \sin x_n$ (c) $x_{n+1} = \tan x_n$ For $10$ bonus ...
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1answer
38 views

Cholesky decomposition with unit diagonal

Let $A$ be real-valued (strictly) positive definite (P.D.) so that it has a unique Cholesky decomposition of the form $A = LL^T$ where $L$ is lower triangular. What PD matrices $A$ have a Cholesky ...
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1answer
62 views

Applied/Numerical Linear Algebra-Suggestions for Project

I am looking for suggestions for a research project in applied/numerical linear algebra. As far as requirements, there really aren't any except that the topic has to tie in somehow with numerical ...
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1answer
50 views

Floating point arithmetic operations when row reducing matrices

A numerical note in my linear algebra text states the following: "In general, the forward phase of row reduction takes much longer than the backward phase. An algorithm for solving a system is usually ...
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2answers
61 views

Explaining roundoff error when row reducing matrices

In my linear algebra textbook (in the context of row reducing and obtaining a matrix in echelon or reduced echelon form), there is a numerical note that reads as follows: "A computer program usually ...
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45 views

Least-squares solution to a transformation between coordinate frames

Suppose I have four coordinate frames in 3D space: A, B, X and ...
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21 views

Mixed Lognormal Model Calibration

Any ideas as to how to calibrate a mixed lognormal volatility model (Brigo and Mercurio 2002) for arbitrary N < 10? The paper seems vague with respect to implementation.
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15 views

LU Decopmostions with block

So both $A_{11}$ and $\hat{A_{22}}$ have $LU$ decompositions say $A_{11}=L_{1}U_{1}$ and $\hat{A_{22}}=L_{2}U_{2}$. Show that $ \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} ...
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78 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} ...
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1answer
27 views

How to prove Kahan's example on componentwise pertubation theory?

In Matrix Computations (4th edition) by Gene H. Golub and Charles F. Van Loan, Problem 3.5.3 asks the following problem (and citing Kahan, William. "Numerical linear algebra." Canadian Math. Bulletin ...
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19 views

Dense Rank Deficient Linear System

What are some of the best methods for solving a Dense Rank Deficient Linear System $Ax = b$, where $A$ is Dense, Symmetric but possibly Rank Deficient. I know SVD can solve it pretty nicely while ...
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1answer
36 views

Perspective transformation matrix application

I need to transform an angled photographed pice of paper to a "flat" image. I found this question & solution here on Mathematics and tried it out for the image given in the solution: The values ...
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129 views

Generalized SVD and weighted SVD

I've the following question: How should I select the $A$,$B$ matrices in the generalized singular value decomposition (GSVD) such that it solves the weighted version of the generalized singular value ...
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1answer
140 views

Convergence of Conjugate Gradient Method for Positive Semi-Definite Matrix

Let $A\in\mathbb{R}^{N\times N}$ be a positive semi-definite matrix, given $b\in\mbox{Col}\left(A\right)$ we want to solve the equation system $Ax=b$ . To add some notation, we define ...
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42 views

Any good books to practice on Endomorphisms as related to Diagonalization, Cayley-Hamilton, etc.?

Well, I am looking for books (graduate level) that covers linear maps (endomorphisms, to be specific) with emphasis on topics related to numerical linear algebra, like: diagonilzation, ...
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42 views

Is matrix multiplication commutative for square matrices representing linear transforms?

On several occasions, I have heard that matrix multiplication is commutative for square matrices $A$ and $B$ when they represent linear transformations. Is this true? I know that in general $AB$ is ...
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55 views

Numerically stable method for angle between 3D vectors

I'm looking for a numerically stable method for computing the angle between two 3D vectors. Which of the following methods ought to be preferred? Method 1: $$ u\times v = ||u|| ||v|| \sin(\theta) ...
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1answer
110 views

Best approach for numerically computing the pseudo-inverse of a covariance matrix

What are the reasons to prefer eigenvalue decomposition over singular value decomposition for numerically computing the pseudo-inverse of a symmetric real matrix? In the case when you want to form the ...