# Tagged Questions

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

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### About the Generalized singular value decomposition (GSVD).

I have studied about Singular value decomposition (SVD) and had solved few numerical examples to understand SVD. Now I am studying Generalized singular value decomposition (GSVD). I followed this ...
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14 views

### Can I detect repeated eigenvalue by inverse iteration?

Suppose all eigenvalues of $A$ are nonnegative. By using inverse iteration $A-\mu I$ for many values of $\mu\ge 0$, I can find eigenvalues of $A$. If $A$ is a $n\times n$ matrix and have different $n$ ...
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36 views

### On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$.  \int_{\triangle}|\nabla ...
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39 views

### Linear algebra vector space [duplicate]

Let $W$ be the set of all solutions $(a, n, b, m)$ of $a + 3b + 4m = 0$, i.e. $$W=\{(a,n,b,m)\in\mathbb{R}^4: a+3b+4m=0; a,n,b,m\in\mathbb{R}\}$$ Show that W is a vector space. Is vector \$(6,8,6,4)\in ...
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### Shifted inverse power method in Octave.

EDIT: Ok, I've managed it. It was very stupid bug... I must write $p=L\(P*z0)$ etc.... I'm trying to write a function which returning vector $a$ (vector of eigenvalues of matrix \$A=A^T \in ...
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8 views

### Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...
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17 views

### Solver for sparse linearly-constrained non-linear least-squares

Reposted from stackoverflow on the advice of Nick Rosencrantz: Are there any algorithms or solvers for solving non-linear least-squares problems where the jacobian is known to always be sparse, and ...
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21 views

### Algorithm to compute similarity computation

I have a similarity transformation of matrices from the type $B = P^{-1}AP$. It is known that $A$ and $P$ are invertible matrices, but not orthogonal. Given that I have the matrices $P$ and $A$ I ...
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### Block matrix notation

Given that $A$ is a real, rectangular matrix of dimension $m \times n$ and $\begin{align} A = \left[\begin{array}{c} I \\ e^{\intercal} \\ -e^{\intercal}\end{array}\right] \end{align}$ is represented ...
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19 views

### How to organise people into groups based on preference of others using matrices?

Okay so a population of n students are asked to provide preferences on who they would like to be grouped with. To do so student's provide numerical ratings on their fellow class mates. The scale can ...
2answers
25 views

### solving linear recurrence - general solution confusion

I've been trying to get my head around this for days. I understand what is going on with the calculation of a linear recurrence and I also understand how the characteristic is obtained. What is ...
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38 views

### Approximation of (FEM) by (FDM)

[Ciarlet 3.4-6] Consider the functional J_h : v = (v_i)\in\mathbb{R}^N\longrightarrow J_h(v)\ :=\ \frac{h}{2}\sum_{i=1}^N\left(\left[\frac{v_{i+1}-v_i}{h}\right]^2 + c_iv_i^2\right) - ...
1answer
52 views

### About iterative refinement to the solution of the linear equations

I want to know what is iterative refinement for improving the solution to the linear equations? How they improve solutions and what are the various techniques for the iterative refinements? Any ...
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20 views

### Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): \$\frac{1}{16} \left[ ...
1answer
50 views

### error for Conjugate gradient method

Suppose A is a real symmetric 805*805 matrix with eigenvalues 1.00, 1.01, 1.02, ... , 8.89,8.99, 9.00 and also 10, 12, 16, 36 . At least how many steps of conjugate gradient iterations must you take ...
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146 views

### Sum of eigenvalues and singular values

How one can prove that for a matrix $A\in \mathbb{C}^{n\times n}$ with eigenvalues $\lambda_i$ and singular values $\sigma_i$, $i=1,\ldots,n$, the following inequality holds:  \sum_{i=1}^n ...
1answer
36 views

### $\lambda_{min}\left (\frac{A+A^*}{2} \right )\leq \sigma_{min}(A)$

For $A \in \mathbb{C}^{n \times n}$, how to show that $\displaystyle \lambda_{min}\left (\frac{A+A^*}{2} \right )\leq \sigma_{min}(A)$?
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57 views

### Minimum eigenvalue and singular value of a square matrix

How to show that the relationship $\left | \lambda_{min} \right | \geq \sigma_{min}$ holds between the minimum eigenvalue and singular value of a square matrix $A \in \mathbb{C}^{n \times n}$?
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### Preconditioning and effects on precision of solution of LSE

In my courses on numerical analysis I have been tought that the main and principle motivation for preconditioning linear systems of equations is to increase the convergence rate of iterative solvers ...
1answer
55 views

### Largest and smallest eigenvalues of a hermitian matrix

How to show that the largest and smallest eigenvalues of a hermitian matrix $A \in \mathbb{C}^{n \times n}$ can be found as: \$\displaystyle \lambda_{max} = ...
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24 views

### What is the error in Newton's Method for Matrix Inversion?

I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds. Suppose we have, as ...
1answer
25 views

### Underdetermined linear systems least squares

I have an underdetermined linear system, with 3 equations and four unknows. I also know an initial guess for these 4 unknows. The article I am reading says: We can solve the system using the least ...
1answer
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### Condition number for non-square matrix?

From what I understand the condition number of a non-square matrix A is its largest singular value divided by its smallest nonzero singular value: $\kappa(A) = \sigma_1/\sigma_n$. Where ...
1answer
29 views

### Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
1answer
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### Polynomial Condition Number

I have a question, from "Applied Numerical Linear Algebra"(James W. Demmel), that I don't know how to do. Consider $\mathbb{R^{d+1}}$ as the set of polynomials of degree $\leq d$ and $S_a$ the set of ...
2answers
43 views

### Matrix Calculus in Least-Square method

In the proof of matrix solution of Least Square Method, I see some matrix calculus, which I have no clue. Can anyone explain to me or recommend me a good link to study this sort of matrix calculus? ...
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37 views

### Singular Value Decomposition uniqueness

I assume $\mathbf{A}$ is an $m\times n$ non-negative real valued matrix such that $m>n$ and $\operatorname{rank}(\mathbf{A})=n$. I have already calculated the overdetermined system for a least ...
1answer
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### Help in Proving a theorem

For the last few days I am trying to prove Result 2 which I have written below that uses the concepts of matrix decompostions to write matrix $A$ in the block form. I need help to prove this ...
1answer
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### Why SVD on $X$ is preferred to eigendecomposition of $XX^\top$ in PCA

In this post J.M. has mentioned that ... In fact, using the SVD to perform PCA makes much better sense numerically than forming the covariance matrix to begin with, since the formation of ...
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22 views

### Boltz method for solving normal equations

Recently I came across an interesting comment in a geodetic paper which follows as: "Initially, the normal equations were solved using the Gaussian method of successive elimination. This method, ...
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### Solve: This System of equations for $X$ (does a real solution, exist?)

How can I solve $AX + diag(X)[I-c]=0$ for $X$? All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...
1answer
47 views

### How to find the Householder transformation?

Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$. I'm sorry that I don't know how to start with this problem. A ...
2answers
54 views

### A question about unitary matrix

We know that a complex square matrix V is unitary if \begin{eqnarray} VV^{*} = V^{*} V = I \end{eqnarray} I want to write matrix V into block matrix form, say $V = [V_1, V_2]$. My question is ...
1answer
36 views

### A basic question about convergence of matrix

I am confused with this very basic question. We know that for a square matrix A the following two properties are equivalent to A being a convergent matrix: 1: \$lim_k\rightarrow \infty \|A^{k}\| = ...
1answer
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### Conjuagate Gradient on Periodic BCs

I'm currently writing a CG solver. It works perfectly fine for Dirichlet boundary conditions, however, I also want it to work with periodic BCs. The problem I'm solving is a 3D Poisson equation. I ...
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32 views

### Determinant error bound is better than norm bound for matrix product

In by textbook on numerical algebra, it states that for a numerical matrix product the error bound: $|A B - \hat{A} \hat{B}| \le c|A| |B|$ is a stronger expression than \$\|A B - \hat{A} ...
1answer
35 views

### what we can say about block inverse besides schur complement

Suppose I have a matrix $M$, which has a block structure $% \begin{bmatrix} A & B \\ B^{T} & C% \end{bmatrix}%$, where A has the inverse. How can I better numerically calculate $A^{-1}B$ ? ...
1answer
29 views

### How do I get a matrix from a coordinate system?

What is the matrix of the reflection at the line $y = x-2$? How do I get the matrix at homogeneous coordinates? I don't get this question at all. I have really no idea of what I am supposed to do ...
1answer
92 views

### Relation between positive definite Hermitian matrices with their inverses

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite. Thanks in advance.
1answer
123 views

### Column space of a matrix?

Recently I started learning about matrices and know for example that the pivot columns of a matrix form a basis for the column space of this matrix. I just can't seem to find out when two matrices ...
2answers
43 views

### Align basis of vector space with that of subspace

Suppose I have two real vector spaces $V,S\subset\mathbb{R}^n$ and $S\subset V$. Say the dimension of $V$ is $l$ and that of $S$ equals $m<l$. They are given in terms of their basis vectors \$v_i, ...
0answers
35 views

### Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
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95 views

### Sum of idempotent matrices is Identity

[Ciarlet, Problem $1.1-10$] Let $A_k$, $1 \leq k\leq m$, be matrices of order $n$ satisfaying $$\sum_{k=1}^mA_k\ =\ I.$$ Show that the following conditions are equivalent. \$A_k = ...
1answer
155 views

### QR Algorithm information

How can we perform 2 iterations of the QR Algorithm to the following matrix? A =\pmatrix{2 & -1 & 0 \\ -1 & -1 & -2 \\ 0 & -2 &3 ...
0answers
40 views

### Criterion for detecting rank-deficiency via QR decomposition?

I apologize in advance if this is an ill-posed question -- I'd appreciate advice on what pieces are missing as much as an answer. I'm solving a system like $P \approx X Y^T$, where P is a large ...
2answers
83 views

### When do two matrices have the same column space?

Recently I started learning about matrices and know for example that the pivot columns of a matrix form a basis for the column space of this matrix. I just can't seem to find out when two matrices ...
0answers
65 views

### Solving Over-Determined Homogeneous Linear Equations with Constraints

I'm trying to numerically "solve" a homogeneous equation of the form $$A \vec{x}=0$$ where $A$ is 2n x 9 ($n\geq3$) and x is a 9D column vector composed of three 3D column vectors stacked on top of ...
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32 views

### maximal m-elements of the matrix inversion

Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
1answer
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### Permutation Matrix

I need to find a Permutation matrix, for an E matrix, i have to permute it because i need to use always two of the rest of eigenvalues of matrix A to operate with them, matrix A is numerically defined ...
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20 views

### Condition number and Chebyshev systems

Suppose I have a square matrix $A$ of size $n$ with elements $a_{mn}=\phi_m(x_n)$ where $\phi_m(x)$ can be thought of as a very friendly function: orthogonal, bandlimited, bounded and analytic. Also, ...
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