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

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1answer
18 views

The equivalence of numerical radius and spectral norm

Let $A$ be a $n\times n$ complex matrix. Define the numerical norm of $A$ as $$w(A)=\sup\{|x^*Ax|;\|x\|_2=1\}, \|x\|_2^2=\sum_{i=1}^n|x_i|^2.$$ And the spectral norm of $A$ is $$\|A\|_\infty ...
0
votes
1answer
8 views

What is and what represents a convergents function in polynomial form?

$$\mathbf{convergents}(cos(1), 20)$$ What exactly is a convergents function and what, that series of fractions is representing ? There is an use for this in numerical linear algebra ? Feel free to ...
0
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0answers
18 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
0
votes
2answers
19 views

Creating a random square matrix with known singular values

The first step in one question has me creating a random square matrix A with singular values given as $2^{-1}, 2^{-2}\dots 2^{-n}$. There is no other information about what assumptions can be made ...
1
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0answers
40 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
1
vote
0answers
8 views

Speed of pseudo-inverse (with possibly ill-conditioned matrices)

I am computing the pseudo-inverse of several matrices of identical size $m \times n$ . However, computation (e.g. with the LAPACK pinv) seems to be much slower in some cases (5 to 10 times slower). ...
-1
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1answer
54 views

Invent a linear mapping given the following conditions. [on hold]

Invent a linear mapping L such that: $L(1,2)=(3,5)$ and $L(-2,1)=(2,-3)$ I'm just unsure on how to start this problem, if anyone can give me any tips that be great thanks!
0
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1answer
23 views

An Inequality in Numerical Optimization

I am reading Jorge Nocedal and Sepher J. Wright's Numerical Optimization and stuck at an exercise 4.6 in chapter 4. The Canchy-Schwarz inequality states that for any vector $u$ and $v$, we have ...
-2
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1answer
34 views

Domain/codmain + range/kernel for linear mappings [on hold]

Consider the linear mapping: $$L(x_1,x_2)=(2x_1-3x_2,4x_1+5x_2,2x_1-x_2)$$ Solve for: (a) Domain and codomain of L (b) Standard matrix of L (c) Basis for the range of L (d) Basis for the kernel ...
2
votes
1answer
47 views

about the power of a matrix

Assume that matrix $A$ contains only 0 or 1 elements. Could anyone give me some condition, under which the matrices $A^i$ (for $i=1,2,3,...,k$) still contains only 0 or 1 elements. For example, I ...
1
vote
1answer
48 views

Solve quadric equation system

How to solve this? For given real and symetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$x^TA_3x=0$$ $$x^TA_4x=0$$
0
votes
1answer
28 views

Finding a non-zero vector in spanA intersection span B using QR factorisation [closed]

Given $A=\{a_{1},a_{2}\}$ and $B = \{b_{1},b_{2}\}$ are linearly independent set of vectors in $\mathbb R^{3}$. I have to device a method to find a non-zero vector that lies in $span(A) \cap span(B)$ ...
8
votes
2answers
243 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
1
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0answers
29 views

Strictly diagonal matrix

Suppose that matrix $A$ is strictly diagonally dominant, show that $||A^{-1}||_{\infty}\leq[min(|a_{ii}|-|\sum_{i\neq j}^n a_{ij}|)]^{-1}$.
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0answers
24 views

Proof the described sequence obey the formula

The sequence start by solving algebraic equation $ P_1V_1^{1.4}=P_2(V_1-\frac{a}{n})^{1.4} $ for P2 and substitute value of $ P_2$ into next equation $P_2(V_1-\frac{a}{n})=P_3(V_1+\frac{a}{n})$. ...
1
vote
1answer
31 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
0
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0answers
28 views

Saddle point problem (KKT) with block-diagonal matrix

Consider the following saddle point problem originating from an interior-point method algorithm: $$ \begin{bmatrix}\mathbf{H} & \mathbf{A}^{T}\\ \mathbf{A} & \mathbf{0} ...
0
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2answers
30 views

Euler method(path s1s2=s2s1)

Given a differential equation $\frac{dy}{dx}=f(x,y(x)), y(x_0)=y_0$. What is the condition for function of f(x,y) such that the result of $y(x_0+S_1+S_2)$ by using Euler forward method, a step size ...
0
votes
1answer
29 views

How do I find transformation matrix with respect to given basis in the domain and/or the codomain, given the transformation in the standard basis?

I´m being given a linear transformation, for which I can find the standard basis in the domain and codomain; but then, the book ask to find the associated matrix related to a new basis for the ...
0
votes
1answer
34 views

An algorithm of solving a non-homogeneous linear equation by random matrices

I'm looking for the proof of the following numerical algorithm. Suppose I want to solve a non-homogeneous linear equation \begin{equation} A x = b \end{equation} The matrix $A$ is non-invertible and ...
0
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1answer
40 views

Is it possible to optimize solution of this linear system?

I have a matrix of the form: ...
1
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2answers
77 views

How to prove the eigenvalues of tridiagonal matrix?

Assume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c & ...
0
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0answers
31 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
1
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2answers
31 views

Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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0answers
19 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...
1
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0answers
24 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
2
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0answers
31 views

Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
0
votes
1answer
10 views

R3 to Planar Subspace Tranform

I'll ask my question three ways to try to maximize my chances of successful communication. I have: a point 'P' in R3 with coordinates $(P_X,P_Y,P_Z)$ A plane Defined by: A point 'O' given by ...
1
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0answers
28 views

Solve linear system with matlab

In my problem, $A$ is a $m \times n$ matrix with $m \geq n$ and $\mathrm{rank}(A)= n$. Let $\Gamma$ be the $(m+n) \times n$ matrix defined by : $$ \Gamma = \begin{bmatrix} A \\ \mathrm{I_{n}} ...
3
votes
1answer
41 views

Generalized inverse/Pseudo Inverse

Let $A_{m. n}$ be a matrix with rank $p$ where $p\leq m$ and $p\leq n$. First Question: We need to show that $A$ can be decomposed as a product of two matrices $A=BC$ where $B$ is an $m$ by $p$ and ...
0
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0answers
20 views

computing leftmost eigenpair of positive-definite matrix

Let $A$ be an $n\times n$ real symmetric positive-definite matrix. Assume that $n$ is large and that $A$ is dense (i.e. it is not sparse). Question: What is the state-of-the-art algorithmically for ...
0
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0answers
38 views

Test for powers method

I have been told that for a normal matrix $A$, the powers method (i.e. computing the succession of Rayleigh quotients for a succession of vectors $z_k=A\cdot z_{k-1}$) can use the following stop ...
3
votes
1answer
44 views

Inverse of a diagonal matrix plus a constant

I am looking for an efficient solution for inverting a matrix in the following form: $D+aP$ where D is a (full-rank) diagonal matrix, a is a constant, and P is a one matrix. This question Inverse of ...
2
votes
2answers
124 views

How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
0
votes
1answer
23 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
0
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0answers
25 views

Are there any sure-fire methods for correctly arranging matricies for Gaussian Elimination?

I am attempting to make a Gaussian Elimination solver for systems of linear equations that contain less than 100 equations. I have roughed out a method for creating and filling in the diagonal of a ...
0
votes
4answers
106 views

Shortest Distance between a Point and a Numerical 2D Curve

I have a 2D Curve. I have all the numerical values for the line within a certain range. I do not have an equation for this line. At several points in this 2D space I want to calculate the shortest ...
0
votes
1answer
27 views

Equation that maps two values to three? [closed]

I'm trying to get an equation for a line in 2-dimensions that maps into an equation for a line in 3-dimensions where $$ f(x,y) => x_1,y_1,z_1 $$ and preferably have a resource that teaches me the ...
0
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0answers
18 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ ...
2
votes
0answers
49 views

compute the bisecting normal hyperplane between two $n$-dimensional points.

I have two points $\mathbf{x_1}$ and $\mathbf{x_2}$, where $\mathbf{x_i}=\{x^i_1, x^i_2, \ldots, x^i_n\}$. I need to find a normal hyperplane $P$ that goes through the midpoint of $\mathbf{x_1}$ and ...
0
votes
1answer
37 views

Jacobi vs. Gauss-Seidel: convergence

I know that for tridiagonal matrices the two iterative methods for linear system solving, the Gauss-Seidel method and the Jacobi one, either both converge or neither converges, and the Gauss-Seidel ...
0
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1answer
31 views

Tridiagonalize matrices with Householder transformation

I know that it is possible to tridiagonalize symmetric matrices by using a Householder trafo. I also found that we can get any matrix to Hessenberg form by using Householder trafos, but I still don't ...
0
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0answers
34 views

Integration of ODE equation in Matlab / Octave

I have a system of 8 ODE's where the initial conditions are in matrix form. $\frac{dT}{dS} = H T$ where T at the initial state is the identity matrix. $T(a) = I$ H is a constant 8x8 matrix T is ...
0
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0answers
16 views

Is the Hessenberg form of a matrix unique?

I have to calculate the Hessenberg form of an matrix using householder reflectors. For real Matrices I get the same result as the 'hess()' function in Matlab, but for Complex Matrices I keep getting ...
0
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0answers
57 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
1
vote
1answer
22 views

Matrices admit a QR decomposition

I just wanted to ask which matrices admit a QR decomposition. I think that all matrices $A \in \mathbb{R}^{m \times n}$ with $m \ge n$ admit a QR decomp. Are these the only ones that have a QR decomp, ...
4
votes
1answer
189 views

Generate arbitrary numerically invertable matrix

I'm designing a unit-test for a matrix inversion function. Currently I make a random matrix as a test case by generating its elements with random numbers uniformly distributed in $[0,1)$. If I ...
1
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0answers
36 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
0
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0answers
53 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
3
votes
1answer
92 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...