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

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1answer
21 views

How do I find a Solution Common to Many Linear Systems?

So I have the following equation: $$ \sum_{n=1}^{N} S_n f_n(x,y,z) = g(x,y,z) $$ And then for every particular set $\xi$ of $N$ random $(x,y,z)$ points, $\forall x,y,z \in {\mathbb R} $, I can ...
1
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1answer
28 views

Calculating $k$ algebraically smallest eigenvalues of a real symmetric matrix

I have a very big matrix assume $1000 \times 1000$. I want to find $k$ of its algebraically smallest eigenvalues where $k$ is $2$ or $3$. I am using MATLAB to solve this problem. My Try: I try to ...
1
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1answer
31 views

Proof for error analysis

I am trying to proof the following equality for matrix error analysis. Sorry for all the syntax. I am new to math stack. Thanks in advance. $$b = Ax$$ $r = A(x-\hat{x})$, where $\hat{x} =$ ...
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1answer
36 views

Elimination problem with polynomial equations involving multiple variables

Hi guys I am very stuck with this problem. I am trying to eliminate 2 out of the three variables it does not matter which one remains, I personally tried keep x and eliminate the others. My question ...
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0answers
11 views

Solving a system of PDE by fixed point Gauss-seidel iteration method

I have the following system of PDE $$ u-u_0=\operatorname{div}\left(\frac{\nabla u-w}{|\nabla u-w|}\right) \tag 1 $$ $$ \frac{w-\nabla u}{|w-\nabla u|} = \operatorname{div}\left(\frac{\nabla ...
-1
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1answer
37 views

Chebyshev interpolation vs equally spaced interpolation [closed]

Which one is better to use and why? What's the advantage of Chebyshev interpolation over equally spaced interpolation and vice versa?
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1answer
66 views

Rayleigh quotients being the diagonal entry of a matrix after orthogonal transformation

So there's this problem in Numerical linear algebra by Trefethan and Bau a textbook I am reading. (It's great! basically taught me MATLAB and some great numerical methods, its also free!) The ...
1
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1answer
28 views

Separable linear programs

Assume, we have two distinct LPs: \begin{equation*} \begin{aligned} & \text{min}_{x_1} & & c_1^Tx_1 \\ & \text{subject to} & & A_1x_1 = b \\ & & & x_1 \geq 0 \\ ...
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1answer
32 views

Under-determined linear system, showing any solution can be written as $x_{0} + Zw$

In my notes for under-determined linear systems, the following is just given as fact, but I've restructured it as a question because I don't quite understand it. Consider $Ax=b$ where $A$ is an ...
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0answers
50 views

Given a CRS stored matrix A, provide an algorithm for calculating vector u.

Given an $NxN$ matrix $A$ and vectors $u,v,b$ such that: $$u_i = {\frac1{a_{ii}}}(b_i - \sum_{j=1,j\neq{i}}^n a_{ij}v_i)$$ And considering $A$ is stored using CRS, provide an algorithm (or ...
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0answers
21 views

number of iterations for the generalized conjugate residuals method?

I have the matrix $n \times n $ defined as: $A=\begin{bmatrix} 0 & 1 & 0 & \dots& 0 \\ 0 & 0 & 1 & \dots &0 \\ \dots &\dots ...
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0answers
23 views

Generalized conjugate residuals method applied to a block matrix

I have the diagonal block matrix A with $2 \times 2$ k-blocks given by : $D_k=\begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix} $. I have to show that the generalized ...
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1answer
35 views

How many divisions needed for LU Decomposition of a square matrix

If a general matrix is of dimensions $n \times n$, how many divisions are needed to compute the LU Decomposition of this matrix? Can we say zero divisions are needed (it can be done with only ...
0
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1answer
64 views

Hyperplane Matrix Linear transformation

Let $$u =\begin{bmatrix}u_1\\.\\.\\.\\u_n\end{bmatrix}$$ be a nonzero vector in $\mathbb R^n$, and let $T:\mathbb R^n \to \mathbb R^n$ be the linear transformation given by $T(x) = u^\top x$. ...
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2answers
37 views

Relation between perturbed matrix and condition number of the matrix

If A is non‐singular but the perturbed matrix (A+δA) is singular, then show that  $$∥A∥/∥δA∥≤y $$ Where y is condition number of the matrix A. Tried for a solution The relation $$(A+δA)(x+δx)=b $$ ...
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0answers
11 views

Power Method: Showing convergence to dominant eigenvector

What follows is taken from Numerical Analysis, by R. Burden and D. Faires: Let $A\in \mathbb{R}^{n\times n}$, with eigenvalues $\lambda_1,\dots,\lambda_n$ such that $|\lambda_1|>|\lambda_2|\ge ...
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0answers
19 views

Conjugate gradient method and rietz values

I'm working on the conjugate gradient method. I have the matrix A, defined as A= diag(v) where $v=[ones(1,10), 11:1000]$. I have to solve the system $Ax=b$ ,b=ones(1000,1) with the conjugate ...
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1answer
49 views

What is the largest floating point number a so that fl(100 + a) = 100?

What is the largest floating point number a so that fl(100 + a) = 100? Here is how float number is computed. $fl(a ⊙ b) = (a ⊙ b)(1 + δ)$. Where $|δ| ≤ ε$. Furthermore, $ε = 2^{-53}$. My ...
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0answers
25 views

Looking for matrices such that $\kappa(A) =1$

Looking clues for this problem. Find all the matrices such that $\kappa(A) = 1$ We define $\kappa(A) = \|A\|\,\|A^{-1}\|$. If I'm looking matrices such that $\kappa(A) = 1$, I was thinking in ...
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0answers
38 views

Find the inverse of $A+uB+vC+uvD+u^2E+v^2F$ where $A,B,C,D,E,F$ are symmetric.

Given scalars $u,v$ s.t. $0<u,v<1$, we seek the properties of the matrix defined by $$P=A+uB+vC+uvD+u^2E+v^2F$$ A is symmetric and positive definite. $B,C,D,E,F$ are symmetric, but might not ...
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1answer
40 views

Symmetric Gauss Seidel iteration

Let $A=L+D+R$, where $L$ is a strict lower triangular, $D$ a diagonal and $R$ an strict upper triangular matrix. Consider the symmetric Gauss Seidel iteration: \begin{align*} ...
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0answers
10 views

How to find How to find ΔU if LU factorization of tridiagonal Matrix A is used to sovle system Ax=b?

How to find ΔU if LU factorization of tridiagonal Matrix A is used to solve system Ax=b? By using forward and back substitution to show that x̂ satisfies ...
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1answer
35 views

What comes after sorting eigenvalues in PCA?

I'm a student, I have to build PCA from scratch using Matlab on iris data. Iris data have 4 features, I want to reduce them to 2. I reached the sorting of eigenvalues step. What is the next step?
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1answer
28 views

Linear Algebra Complex

A homogonous linear system is given by: x1 + x2 = 0 a · x2 + x3 = 0 2·x1 + x2 + a·x3 =0 where a ∈ C, a) Find the determinant of A and give the values of a for which matrix A is ...
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1answer
21 views

How to have a consistent or inconsistent linear algebra equation?

I have been a bit confused about this linear algebra question, if someone can explain it would be great. So my professor is asking us to determine all values of x for which the linear system a, is ...
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0answers
62 views

How to decide if a system is ill conditioned when the matrix condition number is very different for different norms?

A linear system Ax=b is said to be ill-conditioned if the condition number (A)of the coefficient matrix A is far from 1. Consider the system $$\begin{align}x_1 = &b_1 \\ x_1+x_2 = &b_2 \\ ...
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1answer
20 views

Successive over-relaxation vs conjugate gradient

What is the advantages of successive over-relaxation and conjugate gradient methods over each other? When should I use one of them over the other? Here the discussion is limited to solving linear ...
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0answers
19 views

How to find Common (invariant) Subspace between more than two Hankel Matrices?

Note: I am not a mathematician but a control engineer. A general nonlinear $n_{a}^{th}$ order discrete-time state-space model is described by the following equations: \begin{align} ...
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0answers
20 views

Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique.

Prove if a square root of a Herimitian positive semi-definite matrix is also Herimitian positive semi-definite, then it is unique. Here is what I have done so far: By the spectrum theorem, suppose ...
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0answers
25 views

LU Decomposition simplification on a tridiagonal matrix

If I have a tridiagonal matrix that looks like Tn = diag[1, 3, 1], I can do LU Decomposition of it using n - 1 multiplications (by omitting multiplications with 1) but not n - 1 divisions, right? In ...
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0answers
20 views

A problem about the condition number

Given that $I\in R^{n\times n}$ is identity matrix and $||I||=1$.Assumed that Matrix $A\in R^{n\times n}$ is nonsingular,with $\delta A$ satisfying $||A^{-1}||||\delta A||<1$. Then $A+\delta A$ is ...
2
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1answer
19 views

Issue with trigonometry identity related to condition number of matrix

So, in attempting to compute the condition number for the 2-norm of a matrix, I have stumbled upon a problem i can't resolve. I have the formula $$ \frac{1-\cos\left(\frac{n}{n+1} ...
3
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1answer
37 views

partition of block matrices

If $A=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix}$ is a partition of $A$ such that $A_{11}$ and $A_{22}$ are $r × r$ and $(n − r) × (n − r)$ ...
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2answers
39 views

Show that n(n+1)/2 multiplications are required

$a_{11}x_1$+$a_{12}x_2$+$a_{13}x_3$+ ...+ $a_{1,n-1}x_{n-1}$+$a_{1n}x_n$ =$b_1$ $a_{22}x_2$+$a_{23}x_3$+ ...+ $a_{2,n-1}x_{n-1}$+$a_{2n}x_n$ =$b_2$ $a_{33}x_3$+ ...+ $a_{3,n-1}x_{n-1}$+$a_{3n}x_n$ ...
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0answers
15 views

Question related to matrix transformation in sequence spaces.

Let $M=[K_{i,j}]$ be an infinite matrix, where $K_{ij}=1/i \text{ if } 1\leq i \leq j \text{ and } K_{ij}=0 \text{ if } i>j\geq 1$. Then $M$ defines a map $\ell^p \to \ell^r$ iff $p=1 \text{ and } ...
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0answers
13 views

convergnece of QR-method

I'm studying the QR-algorithm. In particular I have this algorithm: For every $k=0..n $ select a shift $\sigma_k$ factorize $A_k-\sigma_k I =Q_kR_k$ multiply $A_{k+1}=R_k Q_k+\sigma_kI$ muliply ...
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1answer
54 views

Guaranteeing Invertibility with Banach Lemma

I'm trying to find an $\epsilon$ for which the Banach Lemma guarantees $I_n + ɛA_n$ is Invertible, where $A_n$ is a matrix of $1$'s, and $I_n$ is the identity matrix, and $n$ can be any dimension. ...
1
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1answer
33 views

Prove that Q is also upper Hessenberg in A = QR

Background: Suppose $\mathbf{A}$ is an $n \times n$ matrix and it is upper Hessenberg. Using QR-factorization, we have $\mathbf{A=QR}$, where $\mathbf{R}$ is an upper triangular matrix and ...
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0answers
116 views

GAXPY Operations

Let A ∈$R^2$, x ∈ $R^k$. Find the first column of M = (A − x1I)(A − x2I)...(A − xkI) using a sequence of GAXPY’s operations. GAXPY: General matrix A multiplied by a vector X plus a vector Y. I tried ...
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0answers
43 views

Compression of a matrix A by V

I can't understand and even can't find any text on Compression of a matrix A by V. meaning if $B=V^*AV$ then B is called the compression of A. What does it mean???
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1answer
40 views

Householder reflection

Let $\tau \in \mathbb C$, $x,y,v \in \mathbb C^n$. I have to show that if i) $|\tau| =\frac{ \|x \|_2}{\|y\|_2}$, ii) $\tau x^H y \in \mathbb R $ iii) $ \rho( x-\tau y)=v$ with ...
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2answers
45 views

Inverse of a block matrix with particular entries

Good afternoon; I have the following block matrix: $X$ = $$\pmatrix{U&M\\M&V}$$ Where $U,V,M$ are square matrices of size $n\times n $, and it holds: $U^2 = V^2 = M^2 = I$ ; with $I$: ...
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3answers
55 views

Matrix Norm Proofs: Dropping the “max” term and denominator

To prove that $||A||_{\infty}≤\sqrt{n}||A||_{2}$, this math.exchange proof does the following: $$||A(x)||_{\infty}≤ ||A(x)||_{2}≤||A||_{2}||x||_{2}≤||A||_{2}\sqrt{n}||x||_{\infty}$$ Given the ...
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0answers
40 views

Solving for intersection of line and a vector function.

How would one approach a problem of finding intersection points between a line $\vec l = \vec S + d \vec t$ and vector of the form $$\vec v = \begin{pmatrix} x \\ y \\f(x, y) \end{pmatrix}$$ I am ...
2
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1answer
38 views

Solving a complex, sparse, linear system using the Schur-complement

Solution method I am repetitively solving sparse linear systems (for the need of ARNOLDI iterations) of the type: $$\underbrace{\begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix}}_J ...
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1answer
44 views

Does an orthonormal matrix preserve the $p$-norm?

Let $\,A\,$ be a $\,n \times k\, $ matrix, and $\,B\,$ a $\,k \times n \,$ be an orthonormal matrix. Is it true that $\,\left\|AB\right\|_p = \left\|A\right\|_p\,$ for every $\,p\neq 2$?
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0answers
27 views

Finding spanning vector sets

Let $V$ be the set of all vectors over the non-negative integers. For any two subsets $S$ and $T$ of $V$, define $S + T$ to include: All vectors in $S$ All vectors in $T$ All vectors that can be ...
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1answer
45 views

Algorithms for computing matrix logarithm.

On my quest to find the holy grail of mathematics become a little bit better at algebra, I have read up on matrix logarithms and exponentials and how useful they can be in investigating groups and ...
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1answer
38 views

Equaity of two norms of matrix

I have to prove that if A is a $n \times k $ matrix and $A^H$ the hermitian matrix of A, $||A||_2=||A^H||_2$. Where $||A||_2=\sup\{ ||Ax||: ||x||\leq 1\}$ and $|| \cdot ||$ is the euclidean nrmm of ...
2
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1answer
42 views

equality of norms

I have to show that $\| A\|_2=\sqrt{\| A^H\times A\|_2}$. $A$ is a $n\times k$ matrix, $\| \cdot \|_2$ is defined as : $$\| A\|_2=\sup\{ \|Ax\|, \|x\|\leq 1 \}$$ and $\| \cdot \|$ is the ...