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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How many digits of of accuracy do I expect the have the solution $x$ of $||Ax-b||=0$

A least-square problem $||Ax-b||=0$ is solved using a backward stable algorithm (In my case, QR decomposition using householder projectors). The condition number is $\kappa(A)=10^5$. If the problem is ...
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Condition number — what is it? [closed]

May someone tell me what the condition number of a matrix is and how it's related to SVD decomposition? I google a lot but got zero results on this.
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Condition value of sparse matrix

I have a sparse matrix really ill-conditionned. I wondered if the places where the non zero values have an impact on the condition value. My matrix is PSD and what I'd like to know is if the condition ...
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Condition number of matrix equation and an $O(\epsilon^2)$ bound

Let $A\in M(n, \mathbb C)$ be an invertible matrix so the condition number is given by $\kappa(A)=||A||||A^{-1}||$ . Suppose $Ax=b=(A+\Delta A)\tilde x$. If $\dfrac {||x-\tilde x||}{||\tilde x||}\le \...
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Absolute condition number for determining the zeroes

Let $f \in C^{n}([a, b])$ and $x_{0} \in$ $(a, b)$ a root of $ f(x)-\alpha=0, \quad \alpha \in \mathbb{R} $ with multiplicity $n$ but not $(n+1)$ Consider a small perturbation $\varepsilon$ of the ...
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Scattering problem involving ill-conditionned linear system

I am studying the scattering of elastic waves and I have to solve the following linear system of equation \begin{align*} v_r^i + v_r^s &= v_r^t \\ v_\theta^i + v_\theta^s &= v_\theta^t \\ v_z^...
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Is differentiation ill-conditioned or not?

EDIT: user14717's comments resolved the conflict described below. The choice of the second norm significantly limits set of perturbations which are considered. The set is reduced from those ...
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Bounding the norm of the inverse matrix

Very strangely, I do struggle with answering the following, seemingly elementary, question. For a square matrix $A$, is there an upper bound for the norm of its inverse? In terms of e.g. the norm of ...
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Upper bound on condition number of row-normalized matrices

I would like to study the condition number of a non-square normalized matrices as function of the original non row-normalized matrix. Let $X \in \mathbb{R}^{a \times b}$ (for $a > b$). We obtain $\...
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Condition inequality in perturbed LS

I have two matrices $A \in \mathcal{M}_{n,d}(\mathbb{R})$ and $B \in \mathcal{M}_{d,d}(\mathbb{R})$ with $B$ being symmetric definite-positive. I am trying to find a condition on $A$ for which I have ...
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Finding a lower bound for the condition number of an arrowhead

Prove that $ Cond_{\infty} (A) \geq C n^2$ for some positive constant C independent of n. $A \in \mathbb{R}^{n \times n}$ so that: $$a_{ij} = \begin{cases} 1 & \text{if } i = 1 \text{ or } j = ...
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The condition number of the matrix

The problem of computing $b$, given $x$, has a condition number $k(A) = ||A||\frac{||x||}{||b||} \le ||A||||A^{-1}||$. It is said that if the second norm is used then the inequality becomes equality ...
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Relation between $\infty$-norm and 2-norm condition number of PD matrix

Consider a positive-definite matrix $A$ in $\mathrm{R}^{n\times n}$ and let $\kappa_{\infty} = \|A\|_{\infty}\|A^{-1}\|_{\infty}$ and $\kappa_2 = \|A\|_2 \|A^{-1}\|_2$, with $\|A\| _p = \sup_{x \ne 0} ...
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How to find the condition number of a 3*3 matrix with singular value 10, 1 and 1/10? [closed]

If the singular values of a 3*3 matrix are 10, 1 and 1/10 respectively, what is the condition number of the matrix? Is such a matrix numerically tractable?
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Restructuring Ill-conditioned problem for better numeric results

I'm not sure if I'm asking in the right place since this is kind of a field overlap but let's see. I have a dynamic equation system and want to optimize a subset of it's parameters. The algorithm ...
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Condition Number and Lipschitz Constant [closed]

I'm interested in mathematical optimization and have been reading about the concepts of Condition Number and Lipschitz Continuity. They seem very related to me. Is the absolute condition number of a ...
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Condition number of a transposed matrix

Does the condition number of matrix $A^T$ equal to the condition number of matrix $A$? The condition number of a matrix is $||A|| \cdot ||A^{-1}||$.
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Prove the following inequalities, condition number.

I am trying to prove the following inequalities: If ${A + \delta A} $ is invertible, prove 1) $$ \frac{|||(A+\delta A)^{-1}-A^{-1}|||}{|||(A+\delta A)^{-1}|||} ={cond}(A) \frac{|||\delta A|||}{\|...
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If $ax^2+bx+c$ has repeated root $x=r$, what is the condition number solving for $r$ if you only change one of $a,b,c$?

I will edit to be specific like Shaun said: If $ax^2+bx+c$ has repeated root $x=r$, what is the condition number solving for $r$ if you only change one of $a,b,c$? The book says that if $r$ is a ...
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Calculate condition number

For $L1$ And $L2$: $L2=D-A(L1+I)^{-1}$ $L1=D-A$ Can we prove that: condition number of $L2$ < condition number of $L1$ ? if yes, how? Where $D$ is an in degree diagonal matrix and $A$ is an ...
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Relative error bound for Ax=b

I have the matrix $A= \begin{pmatrix} 4-\alpha & 12+\alpha \\ 2-\alpha & 6+\alpha \end{pmatrix}$, a pair $\delta x,\delta b\in\mathbb{R^2}\backslash\{0\}$ satisfying $A\delta x=\delta b$, and ...
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Proving $\frac{1}{n}\kappa_2(A)\leq\kappa_1(A)\leq n\kappa_2(A)$

I am trying to show the following inequality is true: $\frac{1}{n}\kappa_2(A)\leq\kappa_1(A)\leq n\kappa_2(A)$. Here we denote $\kappa(A)$ to be the condition number of the matrix $A\in\mathbb{R}^{n\...
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Condition number of a matrix whose submatrices are linearly dependent

Let us define the complex matrix $X \in \mathbb{C}^{2N \times M}$ where $N > M$. Additionally, the matrix ${X}$ consists of the following submatrices: $$ {X} = \left[ \begin{array}[c]. A \\ B\end{...
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Is this matrix well-conditioned?

Take matrix A = \begin{bmatrix}1/3&1/4\\1/4&1/5\end{bmatrix} I have calculated the condition number of A to be 245/3 (using l1-norm) This means we may lose about 2 significant figures of ...
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Condition number of orthogonal matrices induced by the L1 norm

It is easy to prove that condition number for an orthogonal matrix $A$, $\kappa (A) = 1$ when induced by the $L2$ norm using the fact that: $$ ||A||_2 = \text{maximum eigenvalue of }A^TA $$ But how ...
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absolute asymptotic condition number uniqueness differential equation

Suppose that we have a piecewise, differential equation. Say $\frac{dx}{dt} = \begin{cases} x \sin \frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ $x(0) = 0$ I like to ask if the ...
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Estimation of the spectral condition using the Gerschgorin circle theorem

I have been given the following exercise Use the Gerschgorin cirlce theorem to find an upper bound for the spectral condition of a matrix $A$ which is real, symmetric and diagonally dominant. I ...
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$\text{cond}_2^2(M)= \text{cond}_2(M^T M)$ for non square matrix

Let $M \in \mathbb{R}^{m \times n}$ be a matrix with full colum rank. Proof $$ \text{cond}_2^2(M)= \text{cond}_2(M^T M). $$ What I got so far: Denote the pseudo-inverse (Moore–Penrose pseudoinverse)...
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Structure of an ill-conditioned matrix

I have the matrix $$A = \begin{bmatrix} 0.501 &1 & & &\\ &0.502& 1& &\\ & & \ddots & \ddots& \\ & & & 0.509& 1\\ &...
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condition number involving $\cos x$

The function $f(x)=\frac{1-\cos x}{x}$ is to be evaluated at $x\approx 0$. a) Calculate the condition number of $f$ at $x$ and thus find out whether $f$ is well-conditioned or not. b) Write $f$ ...
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Condition number of a diagonal matrix with norm not induced by an inner product

I am trying to prove the following: For any diagonal matrix $D = \mbox{diag}(d_i)$, we have $$\mbox{cond}(D)=\frac{\max|d_i|}{\min|d_i|}$$ The matrix norm is a norm induced by a vector norm, i.e., ...
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Explicit calculation of condition number of a square matrix

According to Wikipedia, the condition number is defined as follows: Assume the linear system of equation $$Ax = b.$$ Let $e$ be the error in b, then the error of the solution $A^{-1}b$ is $A^{-1}e$. ...
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When is the condition number of an orthogonal matrix equal to $1$

If the norm is induced by an inner product, then: $$||A||=\max_{||x||=1}||Ax||=\max_{||x||=1}||x||=1$$ where the second equality holds from the fact that if the norm is induced from an inner product,...
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Inequality regarding condition number of full column rank design matrix.

I am working on a problem to prove the following inequality: $$(1-p_{ii})\kappa^{2} \leq \kappa_{i}^{2} \leq (1-p_{ii})^{-1}\kappa^{2}$$ Where $\kappa = \frac{\sigma_{1}}{\sigma_{p}}$ is the ...
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Upper bound condition number

Let $A$ be a real matrix with all eigenvalues in the interval $(0,1)$. Show that $$\kappa(A)\le\dfrac{2-\lambda_{\min}(A)}{\lambda_{\min}(A)}$$ where $\kappa(A)$ is $A$'s condition number (maximum ...
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Inequality involving condition number

How should I even begin to attempt to show that: $$\frac{\|\bf{x} - \tilde{x} \|}{\|\bf{x}\|} \leq \frac{cond(\bf{A})}{1 - \|\bf{A}^{-1} (\bf{A} - \bf{\tilde{A}}) \|} \left( \frac{\|\bf{b} - \bf{\...
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The condition number of block matrix

Let $$A = \begin{pmatrix} I & B\\ B^H & I\end{pmatrix}$$ with $\|B\|_2 < 1$. Show that $$\|A\|_2 \cdot \|A^{-1}\|_2 = \frac{1 + \|B\|_2}{1 - \|B\|_2}$$ My attempt: I know that $$||A||_2||...
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Absolute and relative condition

Consider the mapping $x \mapsto f(x)$ with $f(x)=1+\|x\|_p$ for $x \in \mathbb{R}_{+}^n \smallsetminus\{0\},$ as $x_i>0$ for $i=1, \ldots, n,$ and $1 \leq p<\infty.$ Compute $\kappa_{\text{abs},...
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Estimating the condition of a symmetric, real matrix using Gershgorin circles

I'm trying to find a way to estimate the condition $\kappa(A)$ of a symmetric, real matrix $A$ based on the norm $\lVert\cdot\rVert_2$ by using the Gershgorin circle theorem. For this, I calculated: $$...
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How to prove the inequality (3.19) in book Numerical Optimization?

I will briefly introduce the problem here. $\theta_k$ is the angle between the chosen descent direction $p_k$ and the steepest descent direction $-\nabla f_k $, that is \begin{equation} \cos \...
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Skeel's condition number vs. classical condition number

Let $A=(a_{ij})_{1 \leq i,j \leq n} \in {R^{n\times n}}$ be a regular matrix and regard the 'classical' condition number $k_\infty(A):=\|A\|_\infty\|A^{-1}\|_\infty$ of the matrix $A$. Now some ...
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Expected value of the condition number of random matrices

I found in this article that $\forall \Gamma$ random matrix with i.i.d. gaussian $(0,\sigma^2)$ entrances $$\mathbb E [\kappa (\Gamma)]=+\infty$$ is that a property of the gaussian distribution or it ...
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What is the intuition behind condition number?

What is the intuition behind defining condition number of the matrix? It seems to that it should have some geometric intuition behind it. But my book fails to explain it I search the web but didn't ...
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Understanding why the inverse looks like this.

This a part from Carl D. Meyer book: But I do not understand why the denominator in (3.8.1) contains $d^{T}c$ and not $c d^{T}$, could anyone explain this for me please?
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For symmetric definite matrix $A$ and $B$, prove that $k(A+B)\le \max(k(A),k(B))$ where $k(X)$ is the condition number of $X$

Here the condition number is defined as the maximum eigenvalue divided by the minimum eigenvalue. For example, if $A=\text{diag}(a_1,a_2),B=\text{diag}(b_1,b_2)$,then $k(A)=\max(\frac {a_1}{a_2},\frac{...
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Condition number linear system

Let $b \in \mathbb R^n$ be fixed. Find the relative condition numbers of the following problem: Find the solution $x \in \mathbb R^n$ of $Ax=b$ for the invertible matrix $A \in \mathbb R^{n\times n}$....
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Condition number changes after the addition on a real block diagonal matrix

Thanks for your visit.🧡 I want to check the change of the condition number between two matrices. There are two real symmetric matrices, one is a real block diagonal matrix $n\times n$, i.e. $$ A = \...
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Condition number of principle submatrix

Suppose we have a $m\times n$ matrix. Consider its principle submatrices (removing the $i$th row and col). How do I find the principle submatrix with the smallest condition number? Currently I loop ...
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Prove this equality about condition numbers $\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$

I am supposed to prove this equality. Let $A$ be an invertible square matrix over $\mathbb R$ $$\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$$ where ${\rm cond}({}\cdot{})_2$ is the condition ...
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Condition number of concatenated matrices

Say I have three matrices $A_1$, $A_2$ and $A_3$ as given below- $A _{1} = \begin{bmatrix} 1 & 5 & 8 \\ 7 & 3 & 4 \end{bmatrix}, \quad A ...