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

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Impossible Schur Factorizations

I am having trouble finding the schur factorization of the following matrix: $A=\begin{pmatrix}3&8 \\ -2&3 \end{pmatrix}$ I followed an algorithm in the book, as well as computing an answer ...
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0answers
37 views

Tridiagonal Gaussian Elimination: Band Storage

I was given this algorithm for Tridiagonal Gaussian Elimination: Band Storage for i = 2:N if W(3,i-1) is zero error('the matrix is singular or pivoting is required') end m = W(4,i)/W(3,i-1) ...
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1answer
115 views

Minimizing the Determinant

I would like to minimize the determinant of the following matrix, det(A) $A = (VV^T+\lambda I)^{-1}$ and $\lambda$ is set to be very small.
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1answer
59 views

Fast Gauss-Seidel convergence on low rank matrices

I stumbled upon the following remarkable fact when experimenting with the Gauss-Seidel iterative solver: First I construct a low-rank symmetric positive semi-definite matrix $A = M^TM$ with M a ...
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1answer
75 views

Prove this matrix is invertible for $n < m-1$

Prove this $(n+1)\times (n+1)$ matrix $\bf{A}$ is invertible for $n < m-1$ and the $x_k$ distinct, \begin{bmatrix} m &\sum_{k=1}^mx_k &\sum_{k=1}^mx_k^2 &\cdots ...
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1answer
66 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
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1answer
61 views

condition number after scaling matrix

Maybe a well-known question. Let $\Sigma$ represent a real symmetric positive definite matrix, i.e. a covariance matrix. Which diagonal matrix $D$ with positive diagonal minimizes the condition ...
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0answers
49 views

Least squares problem where rows are multiplied by a factor

I want to solve the following linear system in least squares sense: $Ax = b$ Where $A$ is a sparse matrix which has more rows than columns. To solve it in least squares sense I would need to solve ...
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1answer
26 views

Proving the equality: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$ where $P=(A-BD^{-1}C)^{-1}$

I am trying to prove the following equality that I need to use as an intermediate step to solve one of my problems. The equality is the following: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$, where ...
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1answer
68 views

Dense symmetric positive definite matrix

How could one define a dense symmetric positive definite matrix (dimension $1000 \times 1000$) with uniformly distributed eigenvalues (with the smallest eigenvalue $1$ and the condition number $100$) ...
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0answers
44 views

linear algebra formulation help

I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by: $$ BE = ...
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1answer
125 views

Help in this exercise about Richardson extrapolation.

We know $F(h)=a_0 +a_1h + a_2 h^3$ $F(1)=4$; $F(1/2)=21/8$; $F(1/4)=145/64$ Find a approximation of $F(0)=a_0$ with Richardson extrapolation method with an absolute error less than $10^{-2}.$ ...
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1answer
27 views

If $λ_i > 0, \forall i$, $A$ is positive definite

Given that $A \in R^{n,n}$, $λ_i $ the eigenvalues and $x_i$ the eigenvectors ($x_i^Tx_j=δ_{ij}$). I have to show that if $λ_i > 0, \forall i$, $A$ is positive definite. My idea is the following: ...
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0answers
74 views

Proof-finding: Power iteration and complexity of the Rayleigh quotient

I'm searching for a proof for this theorem: \begin{align} |\lambda^{(k)}-\lambda_1| = \mathcal{O}\Big(\Big|\frac{\lambda_2}{\lambda_1}\Big|^{2k}\Big) \end{align} where \begin{align} \lambda^{(k)} ...
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4answers
298 views

How to find 2x2 matrix with non zero elements and repeated eigenvalues?

I need to find a 2x2 matrix with non zero elements that has eigenvalue = 1 repeated (double). How can i do that? Thanks!
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0answers
26 views

Doubts on Conjugate and Biconjugate Gradient Method

I am not able to prove that $r^t_iAd_j=0$ for $j\neq i-1$, given $r^t_i$ the $i$-th residual $b-Ax_i$ and $d_j$ the $j$-th $A$-conjugate direction ...
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1answer
87 views

Computing the best-fit plane normal from n points

I've been working steadily through "3D Math Primer for Graphics and Game Development" and am stuck on how the authors derived their equation for the best-fit plane normal given n points. Please note, ...
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0answers
70 views

Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
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2answers
44 views

Can you compute rank r factorization of a n*n matrix in time O(n^2 r)?

I am wondering if you can compute the SVD/eigenvectors of a rank r matrix of size n*n in time O(n^2 r)? My understanding is that standard eigenvector computations involve bringing matrix into ...
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0answers
84 views

Solve system of equations AXB = 0

Is there a common approach to solve a system of linear equations in a form $A^TXB = \bf{0}$? Where $A$ and $B$ are known matrices and $X$ is an unknown matrix. This seems simple enough, there should ...
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2answers
57 views

How to understand or show this?

We have $$F=ABh^{p_1}+\theta (h^{p_2})$$ $$G=Ah^{p_1}+\theta (h^{p_2})$$ We $A$,$B$ are real numbers, $h$ positive, $|h|\leq 1$ , $p_1<p_2$ natural numbers and $\theta(h)$ means that it is of ...
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1answer
55 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
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3answers
93 views

properties of positive definite matrix

If $A$ is a symmetric positive definite matrix can we conclude $A^{n}$ is positive define too? Why? For example for $n=2$: $x^{T}(AA)x=x^{T}(AA^{T})x=(x^{T}A)^2>0$; for $n>2$?
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1answer
40 views

function of matrix and eigen values

I want to calculate exp(A), A is matrix, with numann series. is this series depend of matrix's eigen values? for example if it's eigen values are large, is numann series useful for this function?
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1answer
63 views

Differences between methods for solving linear equation system

I have a huge linear equation system in this form: F=K.Δ as usual form of problems in the finite element method, where the F vector and K are known and Δ vector is unknown. There are several ...
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1answer
162 views

Need matlab help to construct a numerical example for solving system of linear equation for random matrices

I am reading this paper(page 183). In this paper the iterative methods for computing some solution of the general restricted linear equations \begin{eqnarray} Ax = b, ~~~~ x\in R(A^{k})~~~~ b\in ...
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1answer
51 views

most efficient way to find distinct complimenting subspaces over a finite field

Let $V$ be a $n$-dimensional vector spaoce over $\mathbb{F}_p$ and let $W$ be a $k$-dimensional subspace. What's the most efficient way to algorithmically write down a basis for each distinct ...
3
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1answer
82 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
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2answers
53 views

A linear programming to obtain “canonical basis of convex cone”

In my research a I need to solve the linear equation (getting its null space) under some constraints. The matrix is given below: The constraints shall be (x1...x[16]>0, x[17]...x[20] arbitary...) ...
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1answer
53 views

Proof involving Gauss–Seidel method.

I've got a symmetric positive definite matrix $A$ that I decompose into $A=U+R$ where $U$ is the upper triangular portion of $A$ including the diagonal and $R$ is $A-U$. I've shown that $$x^TAx = ...
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1answer
35 views

Find the equation to movement with a middle point locked

I have this scenario: One particle has to go from $0$ point to $y$ point in $1$ sec. The particle needs to start moving at time $0$ ($time=0sec$), and to go accelerating until the middle of time ...
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1answer
71 views

a conjugate gradients result for eigenvalue estimates

Consider the not preconditioned CG-method for a linear system $Ax=b$. Define $\beta_j = \frac{(r_{j+1},r_{j+1})}{(r_j,r_j)}$ and $\alpha_j=\frac{(r_j,r_j)}{(Ap_j,p_j)}$, where $(x,y) = y^Tx$ denotes ...
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1answer
29 views

Conditioning considerations in least square solutions via the normal equations

I'm a little unsure about how to classify conditioning issues with solving least squares equations via the normal equations approach. I'm hoping to get verification that what I say below is correct, ...
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0answers
53 views

Cholesky decomposition using Newton-Raphson

Hi I'm trying to do an alternative algorithm for the Cholesky factorization, which factorizes a symmetric pos. def. matrix $A=R^TR$ where $R$ is upper triangular. I'm curious what happens if you solve ...
3
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1answer
58 views

Solve Ax = b, but I have a function that implements A

I have an overdetermined linear system $Ax = b$. I need to choose an $x$. $x$ has about 100 elements in it. If I had the matrix $A$, I would set x equal $A^\dagger b$, the pseudoinverse of $A$ ...
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1answer
207 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
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0answers
60 views

Solving for A in the system Ax = 0

Consider the system of linear equations $A x = 0$ where $A$ is a $K \times M$ matrix of reals and $x$ is an $M \times 1$ vector of reals. The matrix $A$ is unknown but we can generate $x$s that ...
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0answers
26 views

only calculate diagonal of cholesky decomposition

I have a massive matrix $A$ that I can't hold entirely in memory, but it is possible to easily calculate individual entries ($A(i,j)$). I'm only interested in calculating the diagonal entries of the ...
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0answers
50 views

is 'chasing the bulge' in the implicit QR algorithm exactly the same as reducing a general matrix to hessenberg form?

When performing the implicit QR algorithm, there's a part where you 'chase the bulge' down the diagonal. While it may not necessarily be numerically or computation-time equivalent, is that ...
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0answers
25 views

How to minimize the peak value of this matrix multiplication?

What range or value of $\theta$ will minimize the peak value of $Y $? $$ Y = \begin{bmatrix} 1+j & 2+j & 3+j & 4+j \\ -4-j & -3-j & ...
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0answers
25 views

The least-squares solution for interval data

I would like to solve the least-squares for $\mathbf{Ax} = \mathbf{y}$ with some values in $\mathbf{A}$ and also in $\mathbf{y}$ are interval-valued numbers. In a more detail, e.g.,: $$ ...
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4answers
103 views

Proof $\langle Ax,y\rangle = \langle x,A^*y\rangle$ when $A$ Hermitian

I was trying to understand a proof of why a Hermitian $A$ matrix has its eigenvectors orthogonal. As part of the proof they state $$\langle Ax,y\rangle = \langle x,A^*y\rangle$$ From which property ...
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0answers
132 views

Solving linear system of equations using Successive Over-Relaxation

I was solving a system of linear equations with SOR. I used different values of relaxation factor (w) for the different runs. I found that for all w > w' (1 < w' < 2), the error is the result ...
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0answers
91 views

Orthogonal Procrustes Problem

The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...
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1answer
104 views

Solving Poisson Equation Finite-difference using maple

How do I solving Poisson Equation Finite-difference using maple consider Poisson equation $$\frac{\partial^2u}{\partial x^2} (x,y)+ \frac{\partial^2u}{\partial y^2} (x,y) = x*e^y$$ $0<x<2$ ...
4
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1answer
105 views

QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm ...
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0answers
106 views

when fixed Point Iteration does not converge?

I want to solve a nonlinear system with the fixed point iteration method. I have initial condition,and the answer is known. By using this method the answer converges very slowly about 1000 iteration ...
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1answer
16 views

Are Similar Matrices and Unitary Property related?

Recall that 2 matrices $A, B\in R^{n,n}$ are similar if there exists a matrix $P$ such that $A=P^{-1}BP$. In this case is $P$ always orthogonal?
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1answer
58 views

Find upper Hessenberg by Householder transformation

I have a matrix that looks like this: $$ \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \\ ...
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1answer
50 views

Show that every operator norm is consistent

Is the following a correct way to show that operator norms are consistent? $$ \|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} ...