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} & ...
5
votes
1answer
93 views

Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$ \begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1} $$ where each $d$ is a column of ...
1
vote
1answer
49 views
+50

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
1
vote
1answer
28 views

Finding spectrum with Schur decomposition

let $\alpha$ be a real number and $A_n(\alpha)$ the set of $n\times n$ matrices such that $$\alpha A+A^HA+A^H=I$$ I'd like to find the set of all eigenvalues for the matrices in $A_n(\alpha)$ using ...
1
vote
0answers
20 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
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
2answers
40 views

Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...
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 ...
0
votes
1answer
43 views

Solving 2x2 diagonally dominant matrix systems (non-symmetric)

I have a linear system of the form $Ax=b$ where $A\in \mathbb{R}^{2\times2}, b\in \mathbb{R}^{2\times1}$. A is diagonally dominant and non-symmetric. This is a "kernel" that I am using to solve a ...
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 ...
4
votes
2answers
68 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
1answer
49 views

LU factorisation question

I have a question from a past exam with the solution but i am getting a completely different answer to that of the solution. Could someone please tell me where I am going wrong? question: Find the ...
0
votes
2answers
43 views

Least Squares Solution Confusion

Say if I have an overdetermined system $A\vec x=\vec b$, I can use the normal equations $\implies$ $A^TA\vec x=A^T\vec b$. If I solve for $\vec x$ I will get a "solution" with an error. It says in ...
2
votes
2answers
59 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
1
vote
0answers
28 views

Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
0
votes
0answers
33 views

System of linrar equations and condition number

The relative error of the solution of a system of linear equation $Ax=b$, for any natural norm $\|\cdot\|$ is bounded by $$ \frac{1}{\| A\| \|A^{-1} \|} \frac{\|r\|}{\|b\|} \le \frac{\|e\|}{\|x\|} \le ...
0
votes
0answers
35 views

condition number of orthogonal matrix

Let $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho ...
1
vote
0answers
28 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
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 ...
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 ...
2
votes
0answers
24 views

Cubic convergence of Rayleigh quotient iteration?

Trefethen and Bau, Numerical Linear Algebra, p. 208 states that Rayleigh quotient iteration (combining Rayleigh quotient estimate for eigenvalues and inverse power iteration) converges cubically ...
1
vote
0answers
23 views

If symmetric matrix in a least-square deconvolution problem positive definite?

I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$. $$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$ It is a linear transform ...
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
1answer
76 views

Solving a system of equations using Newton's method

The following paper http://benisrael.net/Newton-MP.pdf provides a way to solve a system of equations using Newton's method. (The theorem begins at the end of page 2) I can't understand the ...
0
votes
0answers
18 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
3
votes
1answer
95 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
3
votes
1answer
215 views

Numerical verification of solution.

I have the non-linear equation \begin{align} &\left( {x}^{2}-1 \right) \left( -\frac{1}{4}\left({\frac { \left( 4\,{x}^{3}+2\, ex \right) ^{2}}{ \left( {x}^{4}+e{x}^{2}+f \right) ...
1
vote
0answers
45 views

Numerical algorithm to solve quadratic eigenvalue problem.

Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum ...
1
vote
1answer
32 views

Differentiating the $QR$ decomposition?

Let $A(t)$ be a smooth family of invertible $n \times n$ matrices with $A(0) = I$, and let $A(t) = Q(t) R(t)$ be the $QR$ decomposition. Given $\dot{A}(0)$, what is an algorithm to compute ...
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 ...
0
votes
1answer
21 views

How to evaluate the accuracy for sparse linear system solver

I'm currently trying to do some experiments on linear solver. However, it's a little hard to get the sense of the numbers. For example, I know large condition number is bad, but how large is bad? ...
1
vote
2answers
59 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: ...
1
vote
1answer
56 views

Is it possible to calculate $e^x$ given $2^x$?

Given a value $x$, if I have a microprocessor instruction that will give me the value of $2^x$, is it possible to calculate (or approximate) the value of $e^x$ ?
0
votes
0answers
42 views

Francis Algorithm (Implicit QR Algorithm)

In Numerical Analysis, we are touching upon QR and Francis Algorithm. I understand that for Francis's Algorithm, we reduce the matrix to its upper Hessenberg form using Householder transform. What I ...
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 ...
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
25 views

Confirming an $LDL^t$ factorization

Hoping someone can confirm my work here. I'm trying to find the $LDL^t$ factorization (http://www.mathworks.com/help/dsp/ref/ldlfactorization.html) of the matrix $$ A = \left(\begin{array}{ccc} 2 ...
2
votes
1answer
103 views

(A + D)x = b … efficiently!

let say we have $A$ to be symmetric positive-definite (SPD), moreover block tridiagonal Toeplitz matrix and $D$ is block diagonal SPD (both with full rank). Let say we know everything about $A$ and ...
2
votes
1answer
41 views

how to find the distinct eigenvectors from a repeated eigenvalue

As the title, how to find the distinct eigenvectors of an certain eigenvalue if the algebraic multiplicity of that eigenvalue is not 1? The target matrix is real and symmetric. I know these ...
0
votes
0answers
41 views

Upper Hessenberg Form

I am given a matrix. I would like to reduce it to its upper Hessenberg Form. We are discussing eigenvalue computations in Numerical Analysis and the textbook just gives the algorithm for it without an ...
3
votes
1answer
53 views

Necessary and Sufficient conditions for convergence of matrix iterations

I need some help figuring out how to go about the iteration part of the problem...I don't really know where to start. If someone can please help take me through it that would be greatly appreciated. ...
0
votes
1answer
53 views

How does LU decomposition work?

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix. My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ ...
1
vote
0answers
25 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
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} ...
1
vote
0answers
22 views

Gaussian Elimination theoretical question

You know how Gaussian Elimination can be broken up into a sequence of L-U premultiplications right? Suppose that there is a matrix $A=a_{i,j} : j=1,...,n$ is an $n × n$ real matrix such that ...
0
votes
0answers
42 views

QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
0
votes
1answer
27 views

Using Givens Rotation on a vector

Say we have a vector v=$[3\ 0\ 4]$. Find a 3x3 orthogonal matrix Q such that only the second component of Qv is nonzero and such that this component is also positive. Is Q unique? I tried ...
0
votes
1answer
69 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
2
votes
0answers
121 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...