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

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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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34 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
22 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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114 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
53 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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25 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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21 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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28 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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23 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
23 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} \pi\right)}{1-\...
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41 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
40 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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17 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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16 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
57 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. $...
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1answer
110 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 $\mathbf{...
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250 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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46 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
43 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 $|p|=\frac{1}{\|x-...
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2answers
48 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
67 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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42 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
49 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
50 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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31 views

Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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29 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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101 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
40 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 ...
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1answer
43 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 ...
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2answers
39 views

A proof of the uniqueness of svd?

I understand the geometric intuition, but the proof by induction in Trefethen book confuses me : it seems to me that a 1*1 complex matrix has infinitely many left and right singular vector pairs? ...
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38 views

Computational methods to minimizing the norm of a matrix monomial.

Linear optimization solves the problem $$\min_{\bf x}\{\|{\bf Ax - b}\|_2^2\}$$ Edit: Some clarification Doing the derivation of the optimum, first expand the norm: $$\|{\bf Ax - b}\|_2^2 = ({\bf ...
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39 views

Rapid calculation of eigenvectors for a submatrix when I know the eigenvectors of the full matrix

I have an $N \times N$ matrix $Z$ that is complex-valued and nonsymmetric in general. I can solve for the eigenvalues and eigenvectors of this matrix numerically. Call the diagonal eigenvalue matrix ...
2
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1answer
63 views

To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a ...
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82 views

Extracting I-Vectors from a GMM-supervector

The probability model for I-vector in GMM-UBM system is : M = M(ubm) + Ty where, M is GMM supervector (means of all gaussian components concatenated in single vector) M(ubm) is UBM supervector I ...
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1answer
48 views

How to substitute and matrix into a functions?

I have $f(x)=2*x_1 +x_2$ how to find $f(m*x)$ if m is a matrix $m=\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}$
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10 views

Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its pseudo-...
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51 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
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24 views

Speeding up Conjugate Gradients iterations for Sparse Matrices?

I've been using Conjugate Gradients to minimize linear systems involving sparse matrices. Although many of my sparse matrices are highly specialized - i.e. for any given row it is easy to know which ...
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55 views

Finite difference: Radial symmetry boundary condition in tridiagonal system?

I am putting together an axisymmetric finite difference solver for Poisson's equation over a non-"rectangular" boundary in axisymmetric cylindrical coordinates. I was planning on using the dynamic ...
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35 views

Error bounds for solution of system of linear equations when coefficients are uncertain

I have a square system $Ax=b$ and would like to know how much the solution $x$ can change when I change the coefficient matrix $A$. I've stumbled upon the condition number, but this seems to apply ...
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1answer
71 views

What mathematics topics pertain more towards applied mathematics?

I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture ...
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1answer
48 views

solve nonlinear system of equation numerically

solve the following system of equations numerically $$2x+2y - e^{xy} = 0$$ $$x^3 + y - xy^3 = 1$$ I'm also asked to solve analytically but I'm pretty sure the closed form solution doesn't exist ...
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1answer
46 views

Improving the performance of eigs for a large spd Problem

I have two large (think around $100.000\times 100.000$), sparse, real symmetric and positive definite matrices $A$ and $B$ and I want to find the smallest generalized eigenvalue $$Ax = \lambda_{\min} ...
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695 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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1answer
33 views

Stability of (floating point) computed variance

Homework Question from Accuracy and Stability of Numerical Algorithms, 2nd Edition, by Nicholas J. Higham, page 33: So every time we store an number and do a operation, we introduce an error bounded ...
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What are some possible reasons for a large condition number?

For this question, please assume that I am talking about the condition number with respect to the spectral norm. That is, $\kappa_2(A) = \|A\|_2\|A^{-1}\|_2 = \frac{\sigma_{max}(A)}{\sigma_{min}(A)}$. ...
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12 views

Proof of QR Algoirthm Convergence

I am reading Trefthen and Bau and the amount of implicit proof steps are killing me. Can someone explain how the statement of convergence for the "pure" QR Eigenvalue Algorithm (Theorem 28.1) is ...
1
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1answer
34 views

how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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37 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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12 views

Linear Relationship of two equations

if $0.46a = 120b$ and $2.68a = 60b$ The relationship is linear. what does $0b$ equal in terms of $a$? what does $1b$ equal in terms of $a$? A method to work this out would also be nice, I have ...