0
votes
0answers
16 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
0
votes
0answers
13 views

Mapping from $\left(-\infty,\infty\right)$ to $[a,b]$ to reduce numerical error

Suppose $A = \left[\begin{array}{cc} \exp\left(x_1\right)&\exp\left(x_2\right)\\ \exp\left(x_3\right)&\exp\left(x_4\right) \end{array} \right]$, where each of $x_i\in\left(-1000,1000\right),$ ...
1
vote
0answers
53 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
4
votes
1answer
175 views

What is the connection between $\rho$ and $\sigma$ if $\rho\rho^T=\sigma\sigma^T$?

I want to prove that there exists a Borel function $R(\rho,\sigma)$ with values in $M^{d\times d}$ defined on $D=\lbrace(\rho,\sigma)\in M^{d\times d}\times M^{d\times d}\,: ...
0
votes
0answers
40 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
2
votes
1answer
38 views

Gauss Seidel Method - How do I avoid calculating $L^{-1}$?

I'm trying to write a matlab code that gets a diagonal dominant matrix $A$, vector $b$, and finds an approximate solution $x$ to $Ax=b$ using Gauss-Seidel Method. I understand the theory. Suppose ...
2
votes
1answer
32 views

A generalization of GMRES

In oder to solve $Ax=b$, GMRES method finds $x_n$ in the $k$-th Krylov subspace i.e.: $$K_n=span\{b,Ab,...,A^{n-1}\}$$ and we have the condition: minimize $\|r_n\|_2$, which $r_n=b-Ax_n$ Now we ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
1
vote
0answers
62 views

Newton's method for multidimensional functions

Can Newton's method be used to find the root of a function f : $\mathbb{R}^n\to\mathbb{R}^m$. Can anyone provide a proof for this? (I have checked the method of solving system of equations with ...
0
votes
0answers
48 views

Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
1
vote
0answers
22 views

Change in Singular Value Decomposition of a matrix on addition of a single row

Given that I know the svd decomposition of a matrix, is there any way to compute the svd decomposition of the matrix obtained by adding a single row to the original matrix? Is there any relation ...
0
votes
1answer
46 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
0
votes
0answers
34 views

Matrix rank when solving using Gram Schmidt

Can I deduce the rank of a matrix (i.e. of size 3x2) when solving an over determined set of equasion using QR Gram Schmidt method? If not, is there a QR-related or other numerical method to find a ...
1
vote
0answers
32 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
1
vote
1answer
45 views

how to find an upper bound of the spectral radius

I've given the real valued matrix $K=K_1+K_2$, with $K_1$ and $K_2$ symmentric and positiv defined. Further there are given this 3 matrices: with $\omega > 0$ and Now I tried the whole day ...
2
votes
1answer
28 views

Gauss-Newton method — is this matrix product invertible?

In the Gauss-Newton method for solving overdetermined systems of equations, the iteration matrix is of the form $(J^t J)^{-1} J^t$, for a $m \times n$ Jacobian matrix $J$ with $m > n$. I was ...
1
vote
1answer
22 views

QR decomposition

Assuming that we have a QR decomposition of a matrix $A \in \mathbb{R}^{m \times n}$, $Q \in O(m)$ and $R \in \mathbb{R}^{m \times n}$ such that $A=QR$, where R is an upper triangular matrix. Now ...
0
votes
1answer
41 views

Backward error for Crout factorization

Ok, can someone please tell me what is the formula for the max error in LU decomposition of Crout factorization?
1
vote
0answers
31 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
2
votes
1answer
46 views

One step Gauss Seidel method

Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ ...
0
votes
0answers
15 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
0
votes
1answer
48 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
1
vote
2answers
65 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
1
vote
2answers
50 views

Square Idempotent matrix: efficient algorithms for finding eigenvectors

Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ ...
0
votes
0answers
47 views

Condition of a matrix proof

I have to show the following inequality: For given two invertible matrices $A,B \in \mathbb{R}^{n\times n}$ show that $k(AB)\leq k(A)k(B)$, where $k(A)=\left \| A \right \|\left \| A^{-1} \right \|$ ...
3
votes
3answers
88 views

Prove that for every vector $V$, $||V||_{\infty} \leq ||V||_2 \leq || V||_1$

$\newcommand{\inf}{||V||_\infty}$ $\newcommand{\two}{||V||_2}$ $\newcommand{\one}{||V||_1}$ Prove that for every vector $V$, $\inf \leq \two \leq \one$ I have tried to look online for a solution to ...
0
votes
0answers
14 views

Singularity check for Homographies

I know that the standard singularity check for a matrix represented in some finite-precision format (IEEE-754 or whatnot) is "the matrix is singular if the reciprocal of the condition number of the ...
2
votes
1answer
167 views

Quick way of finding the eigenvalues and eigenvectors of the matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$

Matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$ has the eigenvalue: $\lambda_i=\alpha-2\cos(i\theta),$ $i=1,\dots,n$ and the corresponding eigenvectors are: ...
1
vote
1answer
57 views

condition number after scaling matrix

Maybe a well-known question. Let $\Sigma$ represent a real symmetric positive definite matrix, i.e. a covariance matrix. Which diagonal matrix $D$ with positive diagonal minimizes the condition ...
4
votes
4answers
270 views

How to find 2x2 matrix with non zero elements and repeated eigenvalues?

I need to find a 2x2 matrix with non zero elements that has eigenvalue = 1 repeated (double). How can i do that? Thanks!
1
vote
0answers
171 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
0
votes
0answers
75 views

A=UL Doolittle method

A - symmetrical positively defined, matrx, 3-diagonal Make a modified Doolittle method A=UL U - upper triangular matrix L - lower triangular matrix with ones on the main diagonal I have to work on ...
0
votes
1answer
60 views

Solving Linear Systems by hand

My professor said for our final we would have to solve linear systems by hand on our final. Some of our questions for interpolation and finding splines involve large 6x6 or 12x12 matrices. What is the ...
1
vote
1answer
194 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
0
votes
0answers
97 views

Tridiagonal system problem with Gaussian elimination and partial pivoting

What happens to the tridiagonal system (shown below) if Gaussian elimination with partial pivoting is used to solve it? In general, what happens to a banded system? $$ \pmatrix{d_1 & c_1 & ...
0
votes
0answers
54 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
2
votes
0answers
80 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
1
vote
1answer
37 views

an estimate for condition number: $\kappa(C^{-1}A)\leq \kappa(C^{-1}B)\kappa(B^{-1}A)$

I'm currently reading through "Domain Decomposition Methods" by Tosseli and Widlund and in the appendix I found the following Theorem: Let A, B, C be symmetric positive definite matrices. Let ...
1
vote
1answer
282 views

Natural Cubic Spline 3 points

I am trying to do a natural cubic spline but I'm having trouble. f(-.0247500)=-.5, f(.3349375)=-.25, f(1.101000)=0 I tried doing the matrix, Ax=b where, h0=h1=.25 an a0=-.0247500, a1=.3349375, ...
0
votes
0answers
303 views

Exact inversion of matrix complexity (by Gaussian elimination)

I would like to check if what I have done is correct. Please, any input is appreciated. Problem statement: Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination ...
0
votes
0answers
33 views

How to generate random matrices when it's singular values are given?

Consider matrix S as nxn diagonal matrix with singular values populated across the diagonals in non-increasing order. I want to know how to create random matrix A whose singular values with be the ...
3
votes
1answer
57 views

Proving an identity

We define $\|x\|_A^2:= x^TAx$ and $(x,y)_M := y^TMx$ for a symmetric positive definite matrix $A$ and an invertible matrix $M$. I want to show the following identity for the errors of Richardson's ...
1
vote
0answers
340 views

Finding error and proving Romberg Integration Method

Let $f$ be a function, which its integral has to be approximated by using romberg method.The $n\cdot m$th cell in romberg matrix (a lower triangular matrix) is given by ...
0
votes
1answer
53 views

Condition Number of Polynomial (Condition Number = 0)

I'm calculating the condition number of a polynomial equation $$ y = (x-2)^{9} $$ for this equation, the Jacobian is equal to ...
1
vote
2answers
78 views

LU decomposition by hand

Can someone show me a step by step solution to calculate the $LU$ decompisition of the following matrix: $A = \begin{bmatrix} 5 & 5 & 10 \\ 2 & 8 & 6 \\ 3 & 6 ...
0
votes
1answer
147 views

Computational efficiency using Gaussian elimination

Assume it took 2 seconds to solve an equation Ax=b for x (where A is a 3×3 matrix and b is a 3×1 matrix) using Gaussian elimination, how much longer would it take to: a) use Gaussian elimination to ...
1
vote
1answer
163 views

How do deal with a giant sparse matrices?

Someone point me in the right direction. I'm looking to do some heavy-duty manipulation of some really large and often very sparse matrices. Naturally, this problem overlaps programming heavily (I ...
1
vote
1answer
133 views

Power iteration provably works if the matrix has a unique eigenvalue $\lambda$ and $\lambda>0$

Let $A$ be a $n\times n$ real matrix and $v_0 \in \mathbb R^n$ s.t. $||v_0|| = 1$. Define a sequence $(v_k)_k$ of $n$-dimensional real vectors by $v_k = A^kv_0 / || A^kv_0 ||$. Assume that $A$ has a ...
1
vote
0answers
220 views

Algebraic ellipsoidal least squares fit

I'm looking to perform a least squares fit in 3D to a quadratic surface of the form: \begin{equation} Ax^2 + Bxy + Cxz + Dy^2 + Eyz + Fz^2 + ax + by + cz + d = 0 \end{equation} by minimizing ...
1
vote
0answers
125 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...