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

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22 views

Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...
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28 views

How to fix error Msg 1801, Level 16, State 3, Line 3 [on hold]

I want add script file in sqlserver2012 but when execute i have error: Msg 1801, Level 16, State 3, Line 3 Please help me Thanks
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15 views

order of convergence for approximations

Let $u \in L^{2}(0,1)$ and $0 < x_{1}< x_{2}<... < x_{n} = 1$, where x$_{k}$ = k$\cdot$h, n$\cdot$h = 1, a partition of the interval [0,1]. Define I$_{k}$(x) = 1 if x $\in$ [x$_{k}$, ...
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0answers
12 views

Location and perturbation of eigenvalues

This is a problem from Horn and Johnson's Matrix Analysis. I'm having trouble showing the bolded parts in the following paragraphs. In fact, I don't really understand what the sentences mean. I would ...
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1answer
27 views

Eigenvalue inequalities for Hermitian matrices

This is a problem from Horn and Johnson's Matrix Analysis. I've tried to follow the problem but I can't find a way to lead to the conclusion the problem is suggesting. Any solutions, hints, or ...
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22 views

A problem on Gersgorin cirle passing through the eigenvalue of an absolute matrix

I'm having trouble solving the following problem. I think I need to show that the matrix $D^{-1}|A|D$ has property SC, but I can't come up with a way to show it. I would really appreciate any ...
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1answer
15 views

A problem about a theorem on irreducible matrix

I'm stuck on a problem where I need to find a counterexample. I'm not sure how to come up with a reducible matrix to show that it doesn't satisfy the result of the following corollary. Any solutions, ...
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1answer
17 views

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$ $\lt C_k'$, for at least one value of $k$

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$$\gt C_k'$, for at least one value of $k=1,\dots, n$, where $C_k'$ denotes $A$'s deleted absolute column sums ($a_{kk}$ is ...
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1answer
18 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
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1answer
15 views

Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$. I'm lost as how to even begin to prove this fact. Any ...
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2answers
29 views

Domain for which this matrix is positive definite

What is the domain for which this matrix is positive definite? $$\left(\begin{array}{cc} 12x^2 & 1 \\ 1 & 2 \\ \end{array}\right)$$ I'm trying to figure this out. I know the ...
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0answers
23 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a symmetric matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
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1answer
50 views

Quality of approximation [closed]

Basically i have written a code in matlab that is used to solve lower triangular system with forward elimination Ax=B. However, the forward elimination script will not give the exact vector but a ...
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1answer
65 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
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0answers
29 views

Is the following matrix Upper Hessenberg?

Does $$ A = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$ properly satisfy the definition of upper Hessenberg?
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32 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
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30 views

Show that Newton’s Method is well-defined for all k and converges to 0 for $x_0>0$

Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for ...
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1answer
18 views

Generalized Eigensystems

I am looking for solution algorithms for a second order generalization of the eigenvalue problem. A, B, and C are n-by-n matrices, I is the n-dimensional identity matrix, $\lambda_i$ is an unknown ...
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16 views

Jacobi Iteration with Shift

The question is to solve a linear system using Jacobi iterations with a shift of mu = 5. My code converges very quickly, but it does not yield the results that MATLAB gives with the backslash ...
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0answers
8 views

Convergence of recursive application of finite-difference operator to $C^{\infty}$ functions

Let $f\colon \mathbb{R}\to \mathbb{R}$ be an arbitrary smooth function (whose extension to a complex differentiable function is entire, if it matters). Let $\mathbf{D}_{h}$ be a finite difference ...
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0answers
24 views

Meaning of singular Jacobian and workarounds to Newton's method

I'm currently working with Galerkin's method to solve differential equations and I have to retrieve unknown coefficients for the truncate expansion. This is just to set the background for why I need ...
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1answer
38 views

Derivative of $\|Ax-b\|_1$

Using least squares approximation $E^2 = \| Ax - b\|^2 = (a_1x - b_1)^2+...+(a_mx-b_m)^2$ The derivative of E^2 at the point $\hat{x}$ is zero if: $(a_1\hat{x}-b_1)a_1+...+(a_m\hat{x}-b_m)a_m=0$ ...
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1answer
31 views

vector space of natural numbers

I wonder, is it possible for the natural numbers (with zero) t be a vector space on SOME field? I understand why it cannot be over real numbers because of muliplication with negative scalar. BUT what ...
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6 views

Get stuck with some statements of convergence rate of the iteration method from “Iterative methods for sparse linear systems (2nd edition) ”

Here are the statements I get from the book and the two highlight parts are what I can not understand well. The questions are: Why we can conclude that $\rho=\rho(G)$ from"the above analysis"? ...
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20 views

Interpolation question about polinomials?

Let $f(x)=x^n$. show that for each $n$ distinct point $x_0,... x_n$ we have $f[x_0,...x_n]=0$ and $f[x_0,...,x_{n-1}]=\sum_{i=0}^{n-1}x_i$. also show that if $x_{i+1}-x_i$ is fixed for each $i\ge 0$ ...
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0answers
17 views

LU Decomposition for the solution of two linear systems

Let's say I have the following linear system: \begin{equation} \left[ \begin{array}{cccc} S&&L^{T}&&A^{T}&&0\\ L&&0&&0&&0\\ ...
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10 views

Applying perturbed matrix to unperturbed eigenvector

Suppose we've got a matrix $P$ and a perturbed version $\hat{P}=P+E.$ Given that $v$ is an eigenvector of $P$ with $Pv=0,$ I'd like to get as sharp a bound as possible on $\hat{P}v$ (in terms of ...
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1answer
23 views

convolution on 2 by 2 matrices

Let $m$ be a positive integer, and let $A_1,B_1 \in \operatorname{SL}(2,\mathbb{Z})$. Can one always find matrices $A_2,B_2 \in \operatorname{SL}(2,\mathbb{Z})$ such that $$ A_1 \left( ...
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33 views

Matlab project - Jacobi method for tridiagonal matrices…

I have to do a project in Matlab to my University and I don't quite understand what I should do. I was given script that solves systems of equations with Jacobi's method with given tolerance and ...
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16 views

$Tn(x) =\frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2}$, show that $|Tn(x)| \le 1,\forall x \in [-1,1]$

$Tn(x) =\frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2}$, show that $|Tn(x)| \le 1,\forall x \in [-1,1]$
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11 views

Consistently obtaining the negative of the correct eigenvalue in hand calculations

The question is "find the eigenvector of A corresponding to the given eigenvalue" A = | 1 0 2 | |-1 1 1 | | 2 0 1 | Eigenvalue = -1 Add -1 * I to A Row reduced to | 1 0 1 | | 0 1 1 ...
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2answers
34 views

A function to convert a vector to a number and vise versa?

Sorry in advance if I didn't choose the right tags for the question, I wasn't sure. So I'm a programmer and writing a saving/loading system for data. The way I was serializing (saving) vectors is via ...
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0answers
30 views

QR Algorithm without Shifts (Trefethen and Bau)

A real symmetric matrix $A$ has eigenvalue 1 of multiplicity 8, while all the rest of the eigenvalues are $\leq 0.1$ in absolute value. Describe an algorithm for finding an orthonormal basis of the ...
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1answer
23 views

Enciphered Message with linear enciphering function.

My semester tests are coming up and as I was looking through past papers I came across this question. I was missing a lot during the beginning of the year and this was no doubt covered during my ...
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3answers
29 views

Divergence of fixed-point iteration for real starting values

Consider the linear system of equations $Ax = b$ with invertible $A\in \mathrm{GL}(n,\mathbb R)$ and $b\in\mathbb R^n$. For $A = M - N$ with invertible $M$ the solution $x_* = A^{-1}b$ is a fixed ...
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1answer
36 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
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17 views

Strictly diagonally dominant matrix -LU factorization

Let $A\in\mathbb{C^{n\times n}}$ be strictly diagonally dominant. I want to show that the LU factorizations with and without partial pivoting are the same for these matrices. For start, I created ...
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1answer
47 views

Why base vectors are 1 in length? [closed]

I can't really find any substantial reference in the math literature that justifies the fact that basis vectors are usually $\begin{bmatrix}1&0&0\end{bmatrix}$ or ...
2
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1answer
45 views

Is there a function in MATLAB that will estimate the initial condition from a set of data?

I was given a state-space model of a system and a list of outputs for t=0 to t=5, sampled every 0.1 seconds and asked to approximate the initial condition. Is there a function in MATLAB that will take ...
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0answers
11 views

Pairing Two Point Clouds

So I have two point clouds $X$ and $Y$ each with $N$ points in the familiar $\mathbb{R}^3$ euclidian 3D space. I then have an inter-point distance $d(\vec x_i,\vec y_j)$ which is zero if $\vec x_i$ is ...
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2answers
37 views

stuck with some parts of the proof about “ matrix is normal iff each of its eigenvectors is also an eigenvector of its transpose conjugate matrix”

When I read the book Iterative Methods for Sparse Linear Systems, Second Edition, I get stuck with the following proof. The yellow highlight parts are the positions I have trouble to understand. ...
2
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1answer
23 views

how to prove the relationship about spectral radius, numerical radius and matrix two norm?

When I read page 24 in Iterative Methods for Sparse Linear Systems, Second Edition, I can not understand the following statement: (My major is not math) Let $A$ be an n-square complex matrix with ...
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0answers
9 views

Relative Error with Respect to Frobenius Norm

I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary. This is a self study, but I believe my question is broad enough so that it can be of ...
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9 views

Dimensionality Reduction

Let $X\in\mathbb{R}^{100\times 100}$ matrix and let its eigenvector and eigenvalues be $X_{vec}$ and $X_{val}$ respectively. If the rank of $X$ is $5$, then is it possible to approximate $X$ with ...
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2answers
43 views

Is $\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1$? For any vector $v\in \mathbb{R}^{n}$

I am stuck while showing that $$\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $v\in \mathbb{R}^n$, and $\|.\|$ is a matrix norm. Here is my steps: I used Frobenius norm: A Frobenius matrix ...
0
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1answer
43 views

How does the Simplex method of solving LPs use the starting solution?

Say one looks at the LP (in slack form) and sees that assigning $0$s to all the non-basic variables doesn't give a valid solution but some other non-trivial assignment of values to the non-basic ...
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2answers
19 views

What does it mean by one matrix is **unitarily similar** to another?

I am reading a tutorial about the Lanczos method for eigen problem / SVD. It mentioned "Then the tridiagonal matrix $B^∗B$ is unitarily similar to $A^∗A$. " What does it mean? I can derive this: ...
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1answer
23 views

conjugate gradient method for semi definite case

Show for a symmetric, positiv semi definite matrix $A$, a vector $b\in Ran(A)$ and initial vector $x_0$: (1) All directions $d_0,d_1,...,d_m$ of the conjugate gradient method are in the range ...
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2answers
48 views

Definition of Spectral Radius / Eigenvalues of Product of a Matrix and its Complex Conjugate Transpose

For any matrix $\textbf{A}$, I know that in the Euclidean L2 induced/operator norm, $\|\textbf{A}\|=\sqrt{\rho(\textbf{A}^{*}\textbf{A})}$, with * being the complex conjugate transpose and $\rho$ ...
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0answers
18 views

Condition number perturbation

I have a matrix of the form $\tilde{H} = H + i A A^\dagger$. It is known that $H$ is hermitian and that $\tilde{H}$ is invertible and $A A^\dagger$ has a kernel of dimension $\geq 1$. I want to study ...