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

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3answers
42 views

upper bound of a function $n^{1/\log(n)}$

I have the following expression $n^{1/\log(n)}, \quad where \quad n \in [1, 10,000]$. When I solve this numericall, I get the resultant value 2.718282 for all $n \in [2, 10,000]$. On this basis, I can ...
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1answer
45 views

An exercise with linear maps

I'm solving the following exercise: $\newcommand{\RR}{\mathbb{R}}$ Let $f: \RR^3 \rightarrow \RR^4: (a,b,c) \mapsto (a+2b+c, b+c, a + 2b + 3c, a + b + 2c)$ Find the basis of $f^{-1}(E)$ ...
1
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1answer
36 views

Choosing value of ω for SOR

I am learning about successive overrelaxation, and I'm wondering if there is an intuitive reason as to why ω must be between 0 and 2. I know that the method will not converge is ω is not on this ...
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0answers
52 views

Dynamically equivalent of numerical solution of $y' = f(y)$.

Consider an autonomous system $y' = f(y)$ and a fixed step size $h$. a) Show that the trapezoidal method applied $N$ times is equivalent to applying first half a step of forward Euler, (i.e forward ...
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0answers
12 views

matrix optimization problem techniques

I'm looking for some resources on learning techniques commonly used in matrix optimization. For example, minimization of the Frobenius/nuclear/weighted norm of a function of a matrix subject to ...
1
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1answer
35 views

Unitary diagonalization of matrices

Can someone tell me whether every square matrix $A\in \mathbb{R}$ unitarily diagonizable? If yes what is a necessary and sufficient condition for a square matrix $A\in \mathbb{R}^{n\times n}$ to be ...
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0answers
54 views

Beginner Linear Algebra 1 equation with 3 variables

I do not understand what to put into the remaining values. I tried to solve for the y and z like I did for the x, but the system is telling me that is incorrect. Some help would be appreciated
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1answer
32 views

Is there any Eigen value decomposition, which can be warm-started?

I have a Matrix, $A$ which is positive semidefinite. No consider, $B=A+\Delta$. I have Eigen decomposition of $A$ and $\Delta_{ij}<= \epsilon_1$, $\Vert \Delta \Vert_F <= \epsilon_2$. Is there ...
3
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1answer
66 views

$LU$ Factorization

Suppose the $A\in\mathbb{R}^{n\times n}$ is nonsingular and that $A = LU$ is its $LU$ factorization. Give an algorithm that can compute, $e_i^TA^{-1}e_j$,i.e., the $(i,j)$ element of $A^{-1}$ in ...
0
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1answer
28 views

Is it true that solving a triangular system using forward or backward substitution numerically stable?

The system is $TX = B$, where $T$ is a triangular matrix, $X$ is a unknown matrix, and $B$ is the RHS matrix. I know the system $Tx = b$ is backward stable where $b$ is a RHS vector. Detail check ...
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1answer
93 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and $\mathbf{...
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0answers
34 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta (...
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0answers
25 views

Computing a new inverse given information from an earlier inverse:

Suppose I have linearly independent set of vectors $v_1 ... v_k \in \mathbb{R}^{k}$. I can let $B = [v_1 ... v_k]$ and by applying Gaussian elimination on the matrix $$ [ B \ I_k] $$ I end up ...
1
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1answer
51 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
1
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1answer
21 views

Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
1
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1answer
44 views

Sparse Matrix or Dense Matrix

My task is to implement the inner product and vector triad forms for a dense $A$ in single and double precision. I have successfully implemented the inner product and vector triad form although, I am ...
2
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1answer
37 views

Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...
1
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1answer
34 views

Order of $LU$ factorisation

Can someone tell me how to calculate the order of a) $LU$ decomposition as well as b)the gaussian elimination of a square matrix $A$? I am at a loss ... Given:: $A$ is a $n\times n$ matrix and ...
2
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1answer
24 views

Finding linearly independent columns of a matrix when $m < n$

I need to a maximal set of linearly independent columns of a matrix $A$. I've googled a lot and seen various solutions, but none of them seem to work for me. What I've seen so far is 1.- Using Cauchy-...
0
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1answer
18 views

Efficient Row Sum of Factorized Matrix

I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix. The matrix was factored using SVD: A -> U, S, V -> U, SxV ...
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1answer
57 views

Matrix-vector product of a banded matrix

Suppose $A\in\mathbb{R}^{n\times n}$ is a banded matrix, i.e., a matrix with all of its nonzero elements on the main diagonal, i.e., $\alpha_{i,i}\neq 0$, the first superdiagonal, i.e., $\alpha_{i,i+1}...
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1answer
61 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 &...
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1answer
32 views

Backward Stability Lemma

Lemma-Let $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$ with components, $\xi_i$ and $\eta_i$, $1\leq i\leq n$, respectively, that are floating point numbers. Computing the inner product $x^Ty$ on a ...
2
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1answer
69 views

In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of ...
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2answers
35 views

Find a spd matrix $C \in \mathbb{R}^{n\times n}$ such that $\langle Cv_i,v_j\rangle = \delta_{ij}$

Let $v_1,\ldots,v_n$ be set of eigenvectors of matrix $A \in \mathbb{R}^{n\times n}$. Find a symmetric positive definte matrix $C \in \mathbb{R}^{n\times n}$ such that $\langle Cv_i,v_j\rangle = \...
0
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1answer
46 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}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{...
0
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2answers
49 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, ...
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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) ≤...
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0answers
42 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
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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 ...
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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) $PP^TE^T=PY^...
0
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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 & c\end{bmatrix}...
0
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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 A^\...
0
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0answers
69 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)} |\...
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1answer
79 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) ...
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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
53 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 (...
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1answer
23 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 ...
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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: $$ \|(I-A)^{-1}\|...
1
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1answer
58 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( \...
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0answers
50 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\\ \end{...
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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
votes
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
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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, \ \text{...
0
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1answer
93 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) - \boldsymbol{1}_{i}^{T}...
2
votes
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 ...