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

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1answer
45 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
0
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2answers
46 views

2-norm of the orthogonal projection

So far, I've deduced that if the rank of A is n, then all the columns of A are linearly independent since A has n columns. As a result, m must be greater than or equal to n. In the case that m = n, ...
0
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1answer
39 views

Prove or disprove if $µ_0(Bx, x) ≤ (Ax, x) ≤ µ_1(Bx, x), ∀x ∈ R^n$, then $κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$

Let $A, B ∈ \mathbb{R}^{n×n}$ symmetric. Show that conditional number $$κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$$ holds, if $B ∈ \mathbb{R}^{n×n}$ is a symmetric positive definite matrix satisfying $$µ_0(Bx, x) ...
0
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0answers
36 views

Hilbert Matrix, Gaussian Elimination with varying pivot strategies, and computation error.

I'm doing a project for my Numerical Analysis class about computational error related to Gaussian elimination, gaussian elimination with partial pivoting, and gaussian elimination with scaled partial ...
2
votes
2answers
65 views

From $Ax=\lambda x$, we have $Ax i = \lambda x i$ , where $i^2=-1$??

Actually I found this problem when I met a question, asking me to prove the eigenvector and eigenvalue of real symmetric matrices are all real. I have already proved the eigenvalue part already, but ...
0
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0answers
23 views

Matrix approximation

How to solve numerically for non-negative full-rank matrices $P$ and $E$ with the following constraints? $Y$ is a known non-negative matrix with $G$ rows and $N$ columns, $G > N$ 1) ...
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2answers
21 views

whether the product of two symmetric matrix with one positive definite is diagonizable

Assume $A$ is symmetric and positive definite, $B$ is symmetric. Proof that $AB$ is diagonalizable and all the eigenvalues are real. I think it is better to write $A=R^TR$ and $B=XDX^{-1}$, but I ...
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0answers
27 views

Cholesky update of $A'A+\gamma I$

Let $A$ be such that $A'A$ is positive definite and admits the Cholesky factorisation $$ A'A = LL' $$ Let us append a column-vector $c$ in $A$ and define $$\bar{A}=\begin{bmatrix}A & ...
0
votes
1answer
39 views

Inequality involving Moore-Penrose pseudoinverse

Let $A=\begin{bmatrix}A_1\\A_2\end{bmatrix}\in \mathbb{F}^{(n+m)\times n}$, with $A_1\in \mathbb{F}^{n\times n}$ and non singular, and $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Show that $$\lVert ...
0
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0answers
63 views

Sum of squares of eigenvalues

Let $\Lambda(A)$ be the sequence of eigenvalues including repeated eigenvalues, if there exist. Show that $$\inf_{X\mbox{ not singular }} \lVert X^{-1}AX\rVert_F^2=\sum_{\lambda\in \Lambda(A)} ...
0
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1answer
72 views

How to determine general form of line equation in 3D from 2 points without using vectors, matrices, etc

For a 2D line equation in General Form ($ax + by + c = 0$) it is possible to calculate all coefficients from two given points as follows: $a = y_1-y_2$ $b = x_2-x_1$ $c = (x_1-x_2) y_1 + (y_2-y_1) ...
0
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2answers
21 views

What does the quadratic form $0.5x^T Ax^T-b^Tx$ find the minimum of?

I'm trying to work through example 2, from here. We start by defining a symmetric positive definite matrix $A$: $\begin{pmatrix} 1.2054 & 0.6593 &1.2299 & 1.2577 & 1.0083\\ 0.6593 ...
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0answers
52 views

LU decomposition with pivot

I'm trying to LU decompose, with pivoting, the following matrix ($A=(a_{ij})$): A = [2 1 2; 1 0 3; 4 -3 -1]; % matlab I cannot make out from my literature ...
0
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1answer
22 views

Strategies for evaluating the action of the solution to a Sylvester equation on a vector

I have a Sylvester equation $AX+XB=C$ with a unique solution. I don't actually need $X$, but rather the matrix-vector product $Xv$ (for some known $v$). It seems most literature concerns itself with ...
1
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3answers
31 views

Matrix norm relation

I've been trying to solve this for 3 hours.. If $A$ is an $n \times n$ matrix with $\|A\|<1$ in any norm, then show that $\|(I-A)^{-1}\| \leq \frac {1}{1-\|A\|}$. My trying is: $$ ...
1
vote
1answer
57 views

Perturbation of roots in Wilkinson's polynomial

I am studying numerical analysis. When I read the online definition I found on this paragraph: Suppose that we perturb a polynomial $p(x) = Π (x−α_j)$ with roots $α_j$ by adding a small multiple ...
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0answers
31 views

Inverted pendelum Matrix numerical derivative

Here I've written a dynamic function as : ...
1
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1answer
30 views

Proof that $ ||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2} $ is normed vector space

I've to prof given $X = \{x_1, x_2, ..., x_n\}$ and function $p$ with property $p(x: X) > 0$ that equation $ ||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2} $ is norm on discrete set $X$. This ...
0
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1answer
36 views

How to generalize C from A and B.

I have Two matrix $A=\left( \begin{array}{ccc} \text a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{array} \right)$ and $B=\left( ...
1
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0answers
48 views

Analytic Bounds for Eigenvalues of a 2x2 Block Matrix

I am trying to find conditions under which all eigenvalues of M will have nonpositive real part (i.e. M is negative semidefinite, I think). $$M = \begin{bmatrix} A & BE^T\\ CE & D\\ ...
1
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0answers
12 views

Numerical algorithm for largest Eigenvalue problem

I am dealing with calculating an eigenvalue problem for differential operator of order 4: $$ \alpha \cdot\Delta^2 u+\Delta u-\Delta(u\cdot u_p(x))=\lambda u $$ where $\alpha\in \mathbb{R}$, $\Delta$ ...
3
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1answer
24 views

How can one show that : $|u_{n+1}-\sqrt{2}|\le\frac{1}{4}|u_n-\sqrt{2}|$

$U_n$ numerical sequence such that : ( For all natural numbers $n$ ) $U_{n+1}=1+\dfrac{1}{1+U_n}$ and $U_0=1$ How can one show that : $|U_{n+1}-\sqrt{2}|\le\frac{1}{4}|U_n-\sqrt{2}|$ I arrived to ...
2
votes
2answers
52 views

linear least squares with equality constraints

I am looking for iterative procedures for solution of the linear least squares problems with equality constraints. That is, my problem is to solve $$\min_{x} \lVert{Ax-b} \rVert _2, \ ...
0
votes
1answer
88 views

What should I know about half vectorization and Kronecker product to do this matrix differentation?

I have a scalar function as follows: \begin{equation*} \ell(\beta, \Sigma, \mu, \Lambda) = \sum_{i=1}^{m} \left[\boldsymbol{y}_{i}^{T} \left(X_{i}\beta + Z_{1} \mu_{i} \right) - ...
2
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0answers
34 views

Computing Cholesky Factorisation by Hand

It is a common exam problem to compute the Cholesky factorisation of a small (typically 4x4) matrix. I know that this can be done by first finding the matrix $U$ in the $LU$-decomposition (e.g. by the ...
1
vote
1answer
26 views

compare complexity of matrix transpose

Given 2 matrices: $X(rows=m,cols=n)$ and $Y(rows=m,cols=1)$, which of the following operations is computationally easy, i.e., easy on the machine? $$X^{T} \times Y \\ or \\ (Y^{T} \times X)^{T} $$
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3answers
49 views

Reconstruct a matrix from given eigenvalues

I wanted to know how can I reconstruct a matrix just from its given eigenvalues. I'm really sorry, cause after working on it for 3 days, I haven't any idea about how to do this, therefore I haven't ...
0
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1answer
31 views

How do I perform Gram-Schmidt on floating point vectors with epsilons in them?

Let $\epsilon$ be a small positive number such that $1+\epsilon$ and $3+2\epsilon$ are machine numbers but $3+2\epsilon + \epsilon^{2}$ is computed to be $3 + 2\epsilon $. Now, let the (classical) ...
0
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0answers
41 views

Way to verify a least-squares solution without actually solving for $x$ and $y$?

I just found the least squares solution of the system $\mathbf{x}A = \mathbf{b} = \begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} 3 & 2 & 1 \\ 2 & 3 & 2\end{pmatrix} = ...
12
votes
2answers
84 views

How to get the SVD of $2AA^T-\operatorname{diag}(AA^T)$ given $A$ and its SVD $A=USV^T$?

Given a matrix $A\in R^{n\times d}$ with $n>d$, and we can have some fast ways to (approximately) calculate the SVD (Singular Value Decomposition) of $A$, saying $A=USV^T$ and $V\in R^{d\times d}$. ...
0
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0answers
43 views

roots of Padé approximating polynomials to the exponential function

I need to obtain (numerically) the roots of the denominator in the Padé approximation to the exponential function $e^{-x}$, in Python. I can calculate its coefficients in closed form (see below). But ...
1
vote
0answers
29 views

Perturbation of a linear homogeneous equation system

Let $A$ be a $n\times(n+1)$ matrix, full row rank. Let $\tilde A=A+\Delta A$ be a perturbation of $A$, again with full row rank. I am interested what is known about bounds on the angle between the ...
2
votes
1answer
51 views

Inverse of a matrix defined by a function

I have a matrix $M$ whose elements are defined by some function $$M_{ij} = f ( |i-j| ) $$ Is it possible to derive a function which defines the elements of the matrix inverse $M^{-1}$ i.e. ...
1
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0answers
16 views

Non-linear ODE with diagonal matrix

I have a differential equation of this form: $\frac{dX}{dr}(r)$= M(r)X(r)$ + (\sum_{i}X_i) D(r)X(r)$ $X(r)$ is a size n vector. $M(r)$ and $D(r)$ are n x n matrices with $D(r)$ diagonal. They are ...
0
votes
1answer
22 views

tridiagonal block matrix

Let us consider a linear system of equations $$ Ax=b $$ Where $A$ is a block tri-diagonal matrix, which is given by $$ \begin{eqnarray} A=\left[\begin{array}{ccccc} A_{11} & A_{12} & \dots ...
1
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0answers
30 views

Existence of Non-Commutative $4 \times 4$ Matrix Multiplication Algorithm

This paper by a Russian gentleman gives an optimal (?) algorithm for $3$ $\times$ $3$ matrix multiplication. It beats a previously known method by reducing the total number of discrete operations from ...
0
votes
1answer
22 views

Splitting Method

Consider the iteration matrix for the general splitting method $M=I-N^{-1}A$ where $N$ is any invertible matrix. Show that if $\lambda =1$ is an eigenvalue of $M$. then $A$ cannot be invertible. I ...
1
vote
1answer
28 views

Change multiple positions of points on circles with different radius

There are some points which are placed on a circular path: Now I want to change the position of some points equals to the distance value(d) respected to their path. I'm using this formula to ...
2
votes
1answer
69 views

Why is the largest eigenvalue Lipschitz continuous and not differentiable?

Let $$ A:\mathbb R^n\to \mathbb R^{nxn} $$ where $A(x)$ is symmetric for any $x=(x_1,..,x_n)$. $$A(x) = A_0+x_1A_1+x_2A_2+...x_nA_n$$ and all $A$ is positive semidefinite. Consider $$ ...
0
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0answers
33 views

Find a symmetric matrix of minimal Frobenius norm

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, And let $$x\in \mathbb{R}^n$$ be such that $\lVert Ax-b\rVert_2 = \min_{z\in \mathbb{R}^n} \lVert Az-b\rVert_2$. Show how to calculate a ...
2
votes
1answer
110 views

Prove Operator Norm is a Norm on linear space [duplicate]

Prove that the operator norm defined by $$\left \| A \right \| = \left \| A \right \|_{V\rightarrow W} = \sup_{0\neq v\in V} \frac{\left \| Av \right \|_{W}}{\left \| v \right \|_{V}}$$ (Given norms ...
0
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1answer
41 views

QR Factorization for Inconsistent Linear System

I am trying to recreate the problem found here on finding the least squares solution to an inconsistent linear system via QR factorization. Can someone explain the part about adding on vectors so that ...
0
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0answers
27 views

Handling large exponents in a matrix

I have four quantities stemming from a 4th order differential equation. I can represent these as a vector which is a product of a 4X4 matrix $$ M=\left\{v,\frac{\partial v}{\partial x},\frac{\partial ...
0
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0answers
19 views

Cutting an Orthonormal Basis

I have constructed an orthonormal basis $\{\mathbf{q_1},\dots,\mathbf{q_n}\}$ for a Krylov set $\mathcal{S}_n(A,\mathbf{x})= \text{span}\{\mathbf{x},A\mathbf{x},\dots,A^{n-1}\mathbf{x}\}$ with ...
0
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0answers
25 views

Relation of the upper triangular factor and the original matrix

Suppose $$PA = LU$$ is the LU factorization(exact) of the square real matrix A, L is the unit lower triangular matrix. Is there a way to determine the relation between the norm of $U$ and the norm ...
0
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0answers
27 views

Why does the Lanczos algorithm make an orthonormal basis for the Krylov subspace?

Starting with a $v_0$= $b_0$=0 and a symmetric positive definite matrix A. Why does the following algorithm forms an orthonormal basis span{$v_1$,$v_2$,...,$v_n$} for $K_n$(A,$v_1$)? for k=1,...,n-1 ...
1
vote
1answer
49 views

Damping iterations

Damping is a way of taming a nonconvergent iteration to get it to converge. Given a splitting matrix $M$, which gives the iteration $$x^{k+1} = x^{k} + M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ the ...
0
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1answer
23 views

equality of the spectrum of two matrices

$Q$ is non singular and A is hermitian. $$V_{+}(A)= Span\{ x: Ax=\lambda x, \lambda > 0 \},$$ $$Q V_{+}(Q^H A Q )= Q Span\{ x: Q^H A Q x=\mu x, \mu > 0 \}.$$ Is it true that $ V_{+}(A)= Q ...
0
votes
1answer
27 views

General form for powers of tridiagonal matrices

Consider a symmetric tridiagonal matrix $A\in \mathbb{R}^{n \times n}$: $$A=\begin{bmatrix} a_1 & b_1 & 0 & \cdots & 0\\ b_1 & a_2 & b_2 && \vdots \\ 0 & ...
0
votes
0answers
60 views

Damped Iteration

For splitting $A = M-P$ a damped iteration with damping factor $\gamma <1$ and scalar $\omega$ is $$x^{k+1} = x^{k} +\gamma M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ $$M =\frac{1}{\omega }I $$ $$P = ...