Questions tagged [condition-number]

Question 1

Problem: Let $A \in \mathbb{R}^{m \times n}$ be a nonsingular matrix. Consider the system $Ax = b$. If $b$ is perturbed by $\delta b$, suppose that the solution of the original system is perturbed by $...

Question 2

Let $A = \begin{bmatrix} 1.0000 & 2.0000\\ 1.0001 & 2.0000 \end{bmatrix}$. Suppose we wish to find $Ax = b$, where $b = (3.0000,3.0001)^T$. Instead of $x$, we obtain $x' = ...

Question 3

I was reading some course and they said that given an eigenvalue $\lambda$, for a matrix A, we can consider an eigenvalue $\lambda'$ for the perturbed matrix $A+\delta A$ such that it's the nearest ...

Question 4

I am given a question that we consider two lines in $\mathbb{R}^2$ given by the equations $y=0$ and $ax+y=b$, with $a,b, \in \mathbb{R}$. We are to compute the intersection point $S$ of the two lines ...

Question 5

The question Consider a linear system of size $10 \times 10$ in the form $\mathbf{D}(\alpha) \underline{R}(\alpha) = \underline{L}(\alpha)$, with $\alpha \in [0,A]$ and $A$ being a large real number. ...

Question 6

Context I am trying to investigate the (inversion) condition number of a Hermitian positive definite matrix when it is diagonally loaded, given by $$ \mathbf{M} = \mathbf{S} + \Delta \mathbf{S} \tag{1}...

Question 7

If anything, it will most likely decrease. Let $A$ be a matrix with $m$ rows and $n$ columns ($m\le n$) and $\sigma_m(A)>0$. Let $A'$ be the same matrix with one row removed. Their condition ...

Question 8

I am struggling with the following problem about singular values from Golub & van Loan's Matrix Computations (4th edition): Problem 5.3.10. Let $A \in \mathbb{R}^{m \times n}$ have rank $n$, and ...

Question 9

Let $J = Q^{-1} A Q$ be the Jordan form of $A$, where $Q$ is a similarity transform matrix. Then, how to minimize the condition number $\kappa(Q) = \|Q^{-1}\| \|Q\|$? Is it a convex optimization ...

Question 10

Suppose I have two matrices, $A, B \in \mathbb R^{n \times n}$, both of which are invertible. Suppose that $A$ and $B$ are $\varepsilon$-close in the spectral norm, i.e. $||A - B||_2 < \varepsilon$....

Question 11

Let $\gamma \in \mathbb N \setminus \{0\}$. I want to find the set of all continuously differentiable functions $f:\mathbb R \to \mathbb R$ such that $f$'s local relative condition number, $\kappa(x)$ ...

Question 12

What is the condition number for an eigenvalue of a generalized Hermitian Eigenvalue problem? So consider the given Generalized Eigenvalue problem \begin{equation} HX=SX\Lambda \end{equation} where $H ...

Question 13

Let $A$ be a symmetric positive definite matrix with an $LDL^T$ decomposition: $$A = L D L^T,$$ where $L$ is a unit lower triangular matrix, $D$ is a diagonal matrix. I would like to prove that the ...

Question 14

Question: I am trying to prove that the condition number of a function $ f(x) = Ax $, where A $\in R^{m\times n}$ and $x \in R^{n\times 1}$ , is given by: $$ \text{cond}(f) = \|A^{-1}\|\|A\| $$ The ...

Question 15

Let $A$ and $B$ be $n \times n$ matrices with $n$ distinct eigenvalues $\lambda_1, \ldots, \lambda_n$ and $\mu_1, \ldots, \mu_n$, respectively. Is it possible to estimate the difference $|\lambda_i - \...

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Questions tagged [condition-number]

Inequality of Perturbation of Linear System [duplicate]

Condition Number of $A$

Proving the eigenvalue conditioning formula

condition number of the intersection point of two lines for infinitesimal perturbations

Complex coefficient matrix with columns reflecting around imaginary axis

Condition number of sum of a Hermitian positive definite and a real diagonal positive matrix

Condition number of a matrix do not increase when data is removed

How to compute the smallest singular value of a block matrix?

How to minimize the condition number of a similarity transform matrix?

Relation between condition numbers and closenenss of inverses of two matrices

The set of all continuously differentiable functions with a constant relative condition number

Condition number of an eigenvalue problem

Prove That the Condition Number of A is Greater Than or Equal to the Condition Number of D in the LDL^T Decomposition

Prove that $\operatorname{cond}(f ) = \|A^{−1}\|\|A\|.$ [closed]

Matrix norm and eigen values

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