# Tagged Questions

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

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### Inverse of generalized arrow matrix $A = M^T * M + I$

If we have the following linear system: Ax=b And matrix A is created by multiplying a rectangular matrix with it's transpose: $A = M^T * M + I$ What is the best method to solve for x for different b ...
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### Proving R is an upper triangular matrix

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. Let $x\in \mathbb{R}^n$ with $\|{x}\|_2=1$ and consider the matrix $P=[x,Ax,\dots,A^{n-1}x]\in \mathbb{R}^{n \times n}$. ...
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### how to calculate spectrum of a large collumn stochastic matrix

Okay, I have a collumn stochastic matrix of order $280\times 280$, the entries are given in an url in some webpage in row format. I need to find all the eigen values and eigen vector corresponding to ...
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### Numerical Methods for finding eigenvalues of large matrices.

I'am writing a small research paper on a problem in linear algebra of my choice. I have chose to do the eigenvalue/vector problem. I know that finding eigenvalues gets pretty much impossible if the ...
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### Fast verification of solution to x'Ax<C

Assume we have some complex vector with N dimensions $\vec x$. We need to verify if this is a valid solution to: $\vec x^HA\vec x<C$ where $A$ is a Hermitian matrix and $C$ is some real constant....
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### Number of operations required for gaussian elimination of tridiagonal matrix

How do I account for (or rather, not account for) the 0's in the matrix so I don't do more operations than necessary? Thanks.
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### Improved error estimate for Conjugate Gradient Method

Let $A \in \mathbb{R}^{n \times n}$ be SPD. The error estimate for the conjugate gradient method is given by \|x_* - x_m \|_A \leq 2 \left( \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1} \...
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### Jacobian of second norm

Find the Jacobian of the following function: (a) $f(x)= \|x -x_0 \|_2$ (b) $f(x)= \log(\|x \|_2)$ Please give me some serious hint!!
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### Let a and $a_1,…,a_m$ be given vectors i n $\mathbb R^n$ .

Show that two statement are equivalent. (a) For all x ≥ 0 , we have $a'x≤ max a'_i􀂂x.$ (b) There exist nonnegative coefficients $b_i$ that sum to 1 and such that $a \le \sum_{i=1}^m b_i a_i$ can ...
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### Maximizing the pairwise Frobenuis distance between M othrogonal matrices

I want to maximize the pairwise Frobenius distance between $M$ orthogonal matrices. That is, I'm looking for $Q_{i}, i = 1, 2, ... M$ such that \begin{equation*} \begin{aligned} & \underset{ 1 \...
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### Derive formula for condition (A)

Consider the linear system $Ax=b$ where $$A = \begin{bmatrix}2&4\\1&2+\varepsilon\end{bmatrix}$$ 1) Derive a formula for $\operatorname{cond}_1 ( A )$, the $1$-norm condition number of $A$....
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### How do I find a Solution Common to Many Linear Systems?

So I have the following equation: $$\sum_{n=1}^{N} S_n f_n(x,y,z) = g(x,y,z)$$ And then for every particular set $\xi$ of $N$ random $(x,y,z)$ points, $\forall x,y,z \in {\mathbb R}$, I can ...
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### Calculating $k$ algebraically smallest eigenvalues of a real symmetric matrix

I have a very big matrix assume $1000 \times 1000$. I want to find $k$ of its algebraically smallest eigenvalues where $k$ is $2$ or $3$. I am using MATLAB to solve this problem. My Try: I try to ...
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### Proof for error analysis

I am trying to proof the following equality for matrix error analysis. Sorry for all the syntax. I am new to math stack. Thanks in advance. $$b = Ax$$ $r = A(x-\hat{x})$, where $\hat{x} =$ ...
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### Elimination problem with polynomial equations involving multiple variables

Hi guys I am very stuck with this problem. I am trying to eliminate 2 out of the three variables it does not matter which one remains, I personally tried keep x and eliminate the others. My question ...
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### Symmetric Gauss Seidel iteration

Let $A=L+D+R$, where $L$ is a strict lower triangular, $D$ a diagonal and $R$ an strict upper triangular matrix. Consider the symmetric Gauss Seidel iteration: \begin{align*} (L+D)x^{(k+1/2)}+Rx^{(k)...