# Questions tagged [condition-number]

The condition number of a matrix is the ratio of the largest to the smallest singular value in the singular value decomposition of a matrix.

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### Uncertainty Principle in Kernel-based Interpolation

If one wants to interpolate or reconstruct a function $f:\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$ on a finite Set $X_n:=\{x_1,\ldots,x_n\}\subset\Omega$ using translates of a ...
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### Show that if $A \mathbf{x} = \mathbf{b}$, then $\text{cond}_p(\mathbf{A}) \geq \frac{\|A\|_p\|x\|_p}{\|b\|_p}$

I have this simple little question to show that the above inequality holds with respect to the p-norm (i.e. the norm defined: by $\|x\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{\frac{1}{p}}$). I ...
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### Quantity reflecting ratio of largest vector element to smallest vector element

Given a vector $v \in \mathbb{R}^n$, is there a well known quantity equal or relating to $$\frac{\max_j \vert{v_j}\vert}{\min_j \vert{v_j}\vert}?$$ I have noted that this quantity is coordinate-...
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### Prove that the condition number of the stiffness matrix in Laplace equation is bounded by $h^{-2}$

I'm a little stuck trying to estimate a condition number in FEM context. I would like to prove the following: Consider the stiffness matrix $K$ for piecewise linear functions on a quasi-uniform mesh ...
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### Adding random noise to an ill-posed problem

Suppose I have found a (finite) solution $x$ to an ill-posed problem, e.g. $$b = A x$$ where $A$ is a $N\times N$ matrix with a large condition number, $b$ is a vector ...
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### Evaluate $f(x)=(1−\cos(x))/x$ for arguments $0<x≪1$

How would one usefully evaluate the function $f(x)=\frac{1-\cos (x)}{x}$ for arguments $0<x \ll 1$ evaluate? Calculate $f\left(10^{-4}\right)$ with an error smaller than $10^{-10}$. Unfortunately, ...
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### Build matrix from existing vectors such that condition number of matrix is minimal

I have a set of $K\gg1$ vectors, each belonging to $\mathbb{R}^N$, where $N\in\mathbb{N}, N> 2$. My goal is to select a subset of $M < K$ vectors from this set of $K$ vectors in order to form a ...
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### Condition number for non-negative least squares

If I want to analyze the stability of a non-negative least squares problem $||Ax-b||_2^2, x\ge0$, how can I measure the stability of the system? If it was a regular least squares problem, then I can ...
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### How to compute the condition number for non-linear systems?

I've been studying numerical methods for solving systems of equations, and I came across the concept of the condition number for linear systems. I understand that the condition number is used to ...
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### Prove that condition number of a matrix increases in its dimension

This is my first time asking a question here. Thus, I apologise in advance if it is not articulated correctly or something else turns out to be wrong with it. Before asking the question itself, ...
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### Minimizing the condition number of matrix-valued function

We have the matrix \begin{equation*}M_a=\begin{pmatrix}a & 1000a \\ 0.5 & 0.5\end{pmatrix}, \qquad a\in \mathbb{R}\setminus\{0\}\end{equation*} I want to calculate the condition number exactly ...
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### Error bounds for perturbation of linear system input

I'm trying to determine an upper bound for the relative error when solving a linear system with a perturbed input. Particularly, I consider a linear system of the form: Ax = b\end{...
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### Condition number of random matrix gets worse as dimension grows

I observed experimentally that, when generating (pseudo) random matrices, their condition number increases with the dimension. Why? When using the norm induced by the Euclidean norm for vector spaces, ...
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### Condition number of subset of columns of a matrix

I have the following question. Let $A \in \mathbb{R}^{n \times m}$, with $n < m$, and let the columns of $A$ have unit $\ell_2$ norm. Say we select a set of indexes $I \subset \{0,1,\dots, m-1\}$ ...
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### Conceptual meaning of eigenvectors and eigenvalues

Can someone provide an intuitive/conceptual explanation for eigenvalues and eigenvectors? I know the mathematical definition $$Ax= \lambda x$$ which means that if x is an eigenvector, then it is only ...
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### Sensitivity in matrix multiplication $C=AB$ - condition number of $A$

Let both $A,B\in\mathbb{R}^{n \times n}$ and positive definite. I know the condition number for $A$ in $Ax=b$ quantifies the sensitivity in this problem. What I am wondering about is if the condition ...
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### On the 2-norm convergence rate of steepest descent method

When minimizing $f(x)=\frac{1}{2}x^THx\quad (H>0)$ , the steepest descent method derives the iteration $$x_{k+1}=x_k-\frac{x_k^TH^2 x_k}{x_k^TH^3 x_k}Hx_k$$. The question is to prove the following ...
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### Upper bound on $\lambda_{max}(V_t)/\lambda_{min}(V_t)$, where $V_t = I + \sum_{i=1}^{t-1} x_i x^T_i$

Consider a matrix $V_t = I_{d\times d} + \sum_{i=1}^{t-1} x_i x^T_i$, where $x_i \in \mathbb{R}^d$ and $\|x_i\|_2 \leq 1$. Is it possible to find upper bound (in terms of $d$ and $t$) for the ratio ...
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### Prove that the condition number $c(A^k) \leq c(A)^k$ for every positive integer $k$ and invertible matrix A. [closed]

I'm not sure where to start here other than $c(A) = \| A \| \|A^{-1} \|$. How does this compare to $c(A^k)$?
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### Condition number of block matrix

Given a block matrix: $$A= \begin{bmatrix} D & E^T \\ E & 0 \end{bmatrix} ,$$ where $D$ is a diagonal, positive entry matrix and $E$ is an incidence matrix of a graph, how is the condition ...
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### Condition number for the sum of two numbers

The condition number for the sum of two numbers $a$ and $b$ is $K=\frac{|a|+|b|}{|a+b|}$. But if I have $a=10$ and $b=-10$, I know that their difference is exactly $0$, so I have in practice no error, ...
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### Cond(PA)=Cond(A)

I want to proof the following equation. A is a nxn regular matrix and P a permutation Matrix $$cond_2(PA)=cond_2(A)$$ So the spectral norm of $||P||=1$ so $cond(PA)=||PA|| \cdot ||(PA)^{-1}||$ I tried ...
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### condition number and singular values

Let $m \geq n$ and $A \in \mathbb{R}^{m\times n}$. Let $\sigma_1,\dots, \sigma_n \geq 0$ be the singular values of $A$. We know that if $r$ is the rank of $A$, then $\sigma_1,\dots, \sigma_r$ are ...
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### How to prove the condition number $\kappa(P) = n$?

I recently have been considering the Sylvester matrix equation given by $AX-XB = C$, where the rank of $C$ is lower than $X$, and $A$ is a diagonal matrix, and $B$ is defined as follow ...
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