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

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3answers
42 views

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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1answer
40 views

The MU-puzzle from GEB

The MUI system only uses the three letters M,U,and I to make strings. The system has four rules that allow you to make new strings out of existing strings by manipulating them. Rules 1 and 2 lengthen ...
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0answers
8 views

Relationship between QR and LU factorization

Both algorithm return very similar results in terms of having a upper/right triangular matrix as one of the factors. What is the relationship between Q and L, and between R and U? What is the ...
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1answer
80 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
10 views

Check whether the quotient that you get after dividing the reciprocal of sum of $-4/15$ and $ 2/5$, by the sum ,is multiplicative identity or not [closed]

Math Check whether the quotient that you get after dividing the reciprocal of sum of $-4/15$ and $ 2/5$,by the sum, is multiplicative identity or not
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2answers
30 views

How to solve the system of equations $\{10^{-4}x_1+x_2=1, x_1+x_2=2\}$ using finite precision arithmetic with three significant figures?

Consider the following two equations: $10^{-4}x_1+x_2=1$ $x_1+x_2=2$ Solve using Gaussian Elimination using finite precision arithmetic with three significant figures. I'm a little ...
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1answer
25 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
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0answers
14 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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0answers
12 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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0answers
23 views

How can I solve this specific set of equations?

Here are the equations: $$\sum_{k = 1}^n i_k + Y_n u_n = J \quad \quad (1)$$ $$i_1 + Y(u_1 - u_2) = J \quad \quad (2)$$ $$i_k - Y(u_{k - 1} -2u_{k} + u_{k + 1}) = 0, \quad \quad k = 2, ..., n - 2 ...
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0answers
20 views

Linear algebra <perhaps an application of Gordan' Theorem>

Question. Let $a_1,...a_n\in\{0,1,-1\}^m$ and $\sum a_i=(1,...,1)$. Is there a permutation $\tau$ of $\{1,...,n\}$ Such that for each $k\in \{1,...,n\}$ the vector $\sum_{i=1}^k a_{\tau (i)}$ has ...
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0answers
25 views

Numerically stable SVD

In this question regarding SVD, it is explained why eigen decomposition of $ A^tA $ is not numerically stable compared to "direct SVD algorithms". Since the former is the algorithm I'm most familiar ...
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0answers
20 views

Overdetermined system with discrete data.

The setup I have a set of experimental data (subscript 1) which calculates two variables $u_1(x,y,z)$ $v_1(x,y,z)$ I can calculate the three spatial gradients for my two variables ($u_1$ and ...
2
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0answers
15 views

Reference request for finite difference method

I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and ...
1
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1answer
29 views

What happens if the power method is applied with a starting vector $q=c_2 v_2+…+c_n v_n$ in the presence of roundoff errors?

Supose $\{v_1,...,v_n\}$ is an eigenvector basis and $|\lambda_1|>|\lambda_2|>\ldots >|\lambda_n|>0$, so, my question is, if our starting vector $q \in span\{v_2,\ldots,v_n\}$ and in the ...
4
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1answer
129 views

QR decomposition help

What do Q and R stand for? Why must the diagonal entries of R be positive instead of just nonzero?
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1answer
32 views

What are some Applications of Hermitian Positive Definite matrices?

I am learning about Hermitian positive definite matrices and the way that they can be decomposed with Cholesky decomposition. I have learned that these matrices deserve special attention for how often ...
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1answer
486 views

Effects of elementary row operation on condition number

How does any elementary row operation on a matrix affect the condition number? Can an ill conditioned matrix be improved by just some elementary row operations? Can I improve the accuracy ...
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1answer
37 views

Iterative method to compute only the positive eigenvalue's and corresponding eignevectors of a very large matrix?

I have a very large dense matrix (~10000 X ~10000) which is not full rank . I want to compute only the positive eigenvalues and corresponding eigenvectors instead of computing all of them. I have ...
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1answer
29 views

How to link the eigenvalues to the components from PCA

I have a difference matrix from daily changes which I use to construct a covariance matrix. On this covariance matrix I use the power method to get the eigenvalues. The power method yields exactly the ...
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0answers
15 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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0answers
24 views

Numerically finding eigenvalues of a Volterra operator of first kind

I'm looking for a solution to the following problem - $\int_{-\infty}^{\infty} K(x-y) f(y) = \lambda f(x)$ Consider $K(x-y) = \left\{ \begin{array}{lr} e^{-(x-y)} & : x > y \\ 0 & : x ...
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0answers
24 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
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1answer
625 views

How to find the Householder transformation?

Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$. I'm sorry that I don't know how to start with this problem. A ...
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1answer
35 views

Solving a system of polynomials in $N$ variables

Suppose I am given some non-negative constants $(C_p)_{p=1, ..., l}$ and I would like to find an integer $N$ and vector $v \in R^N$ such that $$ \sum_{i=1}^N (v_i)^p = C_p $$ for $p=1, ..., l$. Can ...
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0answers
32 views

Eigenvalues after Givens-Rotation

Im just validating my own Code of a Givens-Rotation in Matlab. Therefore i let matlab compute the Eigenvalues after each Givens-Rotation. I am wondering why the Eigenvalues computed by matlab are ...
2
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4answers
58 views

Inverse Chebyshev Recurrence

The Chebyshev polynomials (of the first kind) are a sequence of polynomials defined recursively by $$ \begin{cases} T_{0}(x) = 1 \\ T_{1}(x) = x \\ T_{n}(x) = 2xT_{n-1}(x) - T_{n-2}(x) \end{cases} $$ ...
1
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0answers
33 views

Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
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0answers
26 views

Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
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1answer
29 views

Use the Forward Difference method to approximate the solution to the following PDE?

Use the Forward Difference method to approximate the solution to the following PDE: $$ u^3\frac{\partial u}{\partial t}-x^2u\frac{\partial^2u}{\partial x^2}=2x^8t^7+6x^6t^5+4x^4t^3 $$ for $0\le ...
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0answers
12 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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0answers
52 views

complex rank-one update

I'm trying to find the eigendecomposition of a rank-one update to a complex matrix $D + uv^T$. The matrix $D$ is diagonal, but not the identity. It has unique imaginary entries along the diagonal. ...
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0answers
32 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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2answers
313 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
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2answers
939 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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0answers
31 views

Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay 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, ...
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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, ...
5
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1answer
18 views

Proving a property about Gauss-Seidel

This is a homework problem, so please give hints or tips instead of full answers. The problem is as follows: Let $G$ be the iteration matrix of the Gauss-Seidel method; i.e. $$G=I-(D-L)^{-1} ...
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0answers
5 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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2answers
21 views

Special properties in the direct solving of sparse symmetric linear systems

In the area of computational solving of large sparse linear systems, some solvers specialize only on symmetric sparse matrices, be it positive definite or indefinite as compared to general ...
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0answers
10 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 ...
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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 ...
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2answers
9k views

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I ...
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0answers
24 views

QR Algorithm fails under certain conditions

First of all, i have to admit that i am really knew to this numeric stuff. I have to detect two complex Eigenvalues of a Matrix and therefor i implemented some easy QR-Algorithm with MatLab. I am ...
2
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0answers
31 views

An upper bound for singular-values [closed]

Let $A \in \mathbb{C}^{m\times n}$ and $p=\min \{m, n \}$. Also let $\{\sigma_i\}_{i=1}^p$ be singular-values of $A$ and $\{\tilde{\sigma_i}\}_{i=1}^p$ be singular-values of $A+E$, which $E$ is an ...
1
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1answer
22 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} = ...
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0answers
72 views

Linear Algebra Book like Calculus Made Easy

Now, I know that there are a tons of reference requests for Linear Algebra books but mine is very specific: what is a nice, short, concise, simple, to the point book that gets at the heart of Linear ...
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0answers
28 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
12 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) + ...