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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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Conditioning of covariance after applying element-wise $\tanh$

Define $\gamma:\Re^{n\times n}\to\Re$ that maps a positive definite matrix to the ratio between geometric and arithmetic mean of its eigenvalues: \begin{align} \gamma(M)=\frac{(\det M)^{1/n}}{\frac1n ...
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Find the conditioning of the function

Evaluate the function $f(x)=x+\frac{1}{x}$. Check for what values of $x$ is $f(x)$ ill conditioned. I had this question in a test and using the formula of the condition, $C(x)=|\frac{x f'(x)}{f(x)}|$,...
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Upper bound on $\|\cdot\|_2$ norm of the inverse of a sparse matrix.

Suppose that I have a sparse and invertible $n\times n$ real symmetric matrix $M$ such that $M$ has at most $k$ non-zero entries ($k<<n$) in any row or column. (The case I'm interested in is $k=...
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Prove of matrix condition

The matrix condition number of a matrix $A$ is defined as $\kappa(A) = \Vert A\Vert \Vert A^{-1}\Vert$. I want to show that $$\frac{\Vert x\Vert}{\Vert Ax\Vert}\Vert A\Vert = \kappa(A).$$ Edit There ...
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condition number for a random variable

we know that the condition number measure the maximum relative change, which is attained for some, so, for some random variable $X_i$ we can say that the condition number of the computed $b - \sum_{i=...
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59 views

How to calculate the condition of a function

The terms $a(x)=\dfrac{1-x}{1+2x}-\dfrac{1-2x}{1+x}$ and $b(x)=\dfrac{3x^2}{(1+2x)(1+x)}$ are for $x>0$ the same function $f(x)$. Calculate the condition of $f$ for $0<|x|\ll 1$. How would one ...
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Generate random nonnegative matrix with given condition number

I'm trying to generate random, real, elementwise nonnegative matrices with a given condition number $\kappa$. Dropping the nonnegativity condition I know that this can be done by generating a random ...
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18 views

Max of condition number of a matrix times any stochastic vector

Let $\kappa(A)$ be the condition number of matrix $A$ $$\kappa (A) \equiv \|A^{-1}\|\cdot\|A\|$$ and let $\Delta = \{v \;|\; v_k \geq 0,\forall k\; \text{and}\; \sum_k v_k = 1 \}$ be the set of ...
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20 views

Condition number inequality for non-signular matrices

Let $A \in \mathbb{R}$ be non-singular. Assume that for some induced matrix norm $$\frac{||E||}{||A||} \leq \frac{1}{\kappa(A)}$$ Prove that $A+E$ is non-singular, where $A + E$ is the perturbation of ...
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146 views

Any example of condition number of matrix less than 1? [closed]

Any example of the condition number of matrix less than 1? In our lectures, my professor defined the condition number under the matrix norms. i.e. $$K(A) = ||A^{-1}||_M||A||_M$$
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Find minimal condition number from a non-square matrix

I have an optimization problem for a computational problem and I would like to know how I can solve it. I have a matrix $\left[A\right]_{n \times m}$ with $n < m$ and I would like to get a matrix $\...
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255 views

About condition number

I have the following exercise: Relate the 2-norm condition of $X\in \Bbb R^{m\times n}\ (m\geq n)$ to the 2-norm condition of the matrices: $$B=\begin{equation} \begin{bmatrix} I_m & X\\ 0 & ...
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For a unitary matrix $U,$ prove that $Cond_2(AU) = Cond_2(UA) = Cond_2(A)$

I need help proving this and do not know where to start. Consider a matrix $A \in M_n(\mathbb{C}).$ Prove for any unitary matrix $U, Cond_2(AU) = Cond_2(U A) = Cond_2(A)$
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Condition Numbers

I'm working my way through a basic textbook on Numerical Linear Algebra, and one of the problems asks if the fact that an arbitrary invertible matrix $A \in \mathbb{R}^{m \times m}$ is perfectly ...
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60 views

Why is $\kappa(A) = \infty$ if $A$ is a singular matrix?

Al though not important for my course, I was curious as to why the condition number of a singular matrix is $\infty$. Which also begs the question of what the condition number of a non-invertible non-...
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59 views

Condition numbers of invertible 2x2 matrices

After learning about condition numbers, I worked through some MATLAB examples to compute condition numbers of several 2x2 matrices to gain some intuition. I noticed that for invertible 2x2 matrices, ...
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Why is $\kappa(A^{-1})= \kappa(A)$?

In my syllabus we have the alternative definition of the condition of a matrix: $$\kappa(A)= \frac{\text{max}_{\| \vec{y} \| =1}\| A \vec{y} \|}{\text{min}_{\| \vec{y} \| =1}\| A \vec{y} \|}$$ In it, ...
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Why is $\kappa({A^{-1}})$ important and what does it tell us?

We need the condition number of a matrix $\kappa(A)= \frac{\text{max}_{\vert\vert \vec{y} \vert\vert =1}\vert\vert A \vec{y} \vert\vert}{\text{min}_{\vert\vert \vec{y} \vert\vert =1}\vert\vert A \vec{...
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Matrix condition number analysis

Disclaimer: this is the first exercise of this sort I'm trying to do, so there might be a lot of mistakes. Let $A_{n\times n} = (a_{i,j}) = \cases{1,\quad i=j\\ \theta, \quad i=1,j=n \lor i=n, j=1\\ 0,...
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71 views

How to show that condition number of $QA$ is equal to condition number of $A$

If $A$ is an invertible matrix and $Q$ is an orthonormal matrix, show that $$k(QA) = k(A).$$ Hint: $k(A) = \frac{\sigma_{1}}{\sigma_{n}}$ (the ratio of the largest and smallest eigenvalues).
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What are $\vec{x}$ and $\vec{b}$ in $A \vec{x} = \vec{b}$?

The condition of a matrix gives a bound on how inaccurate the solution $\vec{x}$ will be after approximation. So given $A \vec{x} = \vec{b}$, how much does $\vec{x}$ change when there is a change in $\...
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100 views

Relative Condition number of composite function

I want to find a function $$h = g \circ f$$ such that condition number of $g$ and $f$ are greater than $10$, but the condition number of $h$ is less than $1$. I am trying to use polynomials like $x^{...
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69 views

Condition number of matrix proof

The condition of a differentiable function $f:X\rightarrow Y$ is defined as $$\kappa = \frac{\Vert x\Vert_X}{\Vert f(x)\Vert_Y}\Vert Df(x)\Vert_{\leadsto Y},$$ where $\Vert\cdot\Vert_X$ is the norm ...
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Prove that $\frac{1}{K(B)}\frac{\|A-B\|}{\|A\|} \le \frac{\|A^{-1}-B^{-1}\|}{\|B^{-1}\|} \le K(A)\frac{\|B-A\|}{\|A\|}.$

Let $A$ and $B$ be invertible matrices and $K(C)$ the condition number of an invertible matrix $C$. Prove that $$\frac{1}{K(B)}\frac{\|A-B\|}{\|A\|} \le \frac{\|A^{-1}-B^{-1}\|}{\|B^{-1}\|} \le K(A)\...
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Bound on steepest descent

I'm learning about the method of steepest descent for approximating the solution of $Ax=b$ where A is an invertible matrix. Here is the part I understand: We do this by minimizing the function $f(x)=\...
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Increase numerical stability of a computation

Let $A$ be a $N\times N$ symmetric positive definite real matrix, and $f$ a column vector of length $N$. Moreover, let's call $C$ the inverse of $A$ and $\alpha$ the vector of solution of the linear ...
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append row to achieve minimum condition number

I have a 2-by-3 matrix $A$ with rank 2, rows not orthogonal. What vector $v$ to append such that the resulting matrix $$ \tilde{A}= \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{...
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42 views

Why is an indefinite matrix tough to solve for direct and iterative solvers?

One can sometimes read the rather vague comment that indefinite matrices are "tough to solve" by iterative and direct solvers. An example would be a saddle point system that arises from the ...
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78 views

Condition number for a least square problem

For an $m×n$ matrix A with $m \geq n$. We define $ k(A) = \frac{\max_{\lVert x\rVert=1} \lVert Ax \rVert}{\min_{\lVert x\rVert=1} \lVert Ax \rVert} $ For the euclidean norm, that is, $\lVert A \rVert$ ...
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24 views

Condition number on perturbations cannot cancel $||x||$ in inequality.

Theorem Let $A$ be a square matrix, then $$\lim_{k \to \infty } A_{k} =0 \rightarrow \rho(A)<1$$ Moreover, the geometric series case $\sum_{k=0}^{\infty} A_{k}$ is convergent iff $\rho(A) < 1$. ...
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59 views

For any non-singular $A$, $\frac{1}{\kappa(A)}\leq\frac{\|E\|}{\|A\|}$ if $E+A$ is singular [closed]

Let $A$ be a non singular, and $A+E$ be singular matrix. Prove that $\frac{1}{\kappa(A)}\leq\frac{\|E\|}{\|A\|}$. I thought of assuming $\|A+E\|=0$, but it doesn't seem to be right.
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31 views

Range of values of parameter $\gamma$ in perturbation matrix

I have an orthogonal matrix $A \in \mathbb{R}^{10^{5}\times 10^{5}}$ and a perturbation matrix $\delta A = \epsilon A, \epsilon = 10^{-16}.$How can I calculate the range of values of parameter $\gamma ...
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131 views

Upper bound on condition number in linear preconditioning

I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia: Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
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26 views

Proof or reject condition number if square matrix [closed]

I want to proof(if true) or reject: For all A Square matrix nxn K(10A) = 10K(A), i think it's wrong since: K(cA)=K(A) (refe Show property of condition number) if K(cA) == K(A) which is != from 10K(A), ...
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220 views

Finding the condition number of a matrix $A$, and the values of b such that $A$ is ill-conditioned

Let $A=\begin{pmatrix} 1 & b \\ 0 & 1\\\end{pmatrix}$, what is the condition number of A, for which values of b the matrix A is ill-conditioned. My trial: Since it does not specify in which ...
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Find the condition number and show its instability

If I have $f(a,b)=a+b$ and first I have to find the condition number: I got to $(cond f)(a,b)=\dfrac{a}{a+b}+ \dfrac{b}{a+b}=1$ But the rest of the question says to show the subtraction of close ...
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Upper bound for condition number

I want to show that if $A=I+B$ and $||B||<1$ then $||A||||A^{-1}|| \leq \frac{1+||B||}{1-||B||}$, where $A\in\mathbb{C^{nxn}}$ and $||.||$ is a matrix norm We know that since $||B||<1$, $A$ is ...
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Show that $\min \left\{ J(B) = \frac{\| B \|_2}{\| A \|_2 } \colon \det (A + B) = 0 \right\} = 1 / \mathcal{K}_2(A)$.

Given a real $n \times n$ invertible matrix $A$, show that $$ \min \left\{ J(B) = \frac{\| B \|_2}{\| A \|_2 } \colon \det (A + B) = 0 \right\} = \frac{1}{\mathcal{K}_2(A)}, $$ where $B$ is a real $...
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Condition number and its accuracy

According to the books, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. However, we also knew that - "As ...
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63 views

Badly Conditioned Matrix Error - Should I concern myself with it?

I have a 16 by 16 matrix A with increasing powers in each row, and a vector b. When I try to use Mathematica to solve Ax=b (using LinearSolve) I get an error warning me there may be significant ...
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Find the condition number for $w\left(a,b\right)=a-\sqrt{a^2-b}$ - Can someone check my solution?

$$w\left(a,b\right)=a-\sqrt{a^2-b}$$ $$w\left(\alpha ,\beta \right)=\alpha -\sqrt{\alpha ^2-\beta }=a\left(1+E_a\right)-\sqrt{\left(a\left(1+E_a\right)\right)^2-b\left(1+E_b\right)}$$ $$\frac{\left|w\...
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1answer
72 views

Find the Condition Number for $w\left(a+b\right)=a^2+b^2$

This is what I got: $$w\left(a+b\right)=a^2+b^2$$ $$ \alpha =a\left(1+E_1\right),\:\beta \:=b\left(1+E_2\right)$$ $$ \frac{\left|w\left(\alpha ,\beta \right)-w\left(a,b\right)\right|}{\left|w\left(a,b\...
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what does condition number of not complete power of 10 imply?

so if I have a condition number like $5*10^9$, does this mean that I might lose a upward of 9 significant figures, or 10? I know that if it is $10^9$ then it will lose up to 9 but not sure for ...
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32 views

$\left[\operatorname{cond}(A)\right]^2 = \|{A}\|^2\|\left(A^TA\right)^{-1}\|$ for $m \times n$ matrix $A, m > n.$

I'm following the text on Scientic Computing by Michael Heath and they state $\left[\operatorname{cond}(A)\right]^2 = \|{A}\|^2\|\left(A^TA\right)^{-1}\|.$ I know for rectangular matrices $\...
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43 views

How does lattice reduction algorithm reduce condition number of a matrix?

I've read from a book that "An ill-conditioned matrix can be transformed to a well-conditioned one through Lattice reduction algorithms." The ill-conditioned matrix means that the condition ...
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1answer
44 views

How to Prove that $ cond(A)\ge \frac{||A||}{||C||} $ for any induced matrix norm. With $A$ invertible and $A+C$ singular matrix

How could I prove that for any induced matrix norm $$cond(A)\ge \frac{\|A\|}{\|C\|}$$ where $A$ is an invertible square matrix and $A+C$ is a singular matrix?
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219 views

Norm-wise condition number vs component-wise condition number

I'm going through my lecture notes for a numerical linear algebra class and there were a few things in the chapter which covers condition numbers which I did not quite understand. Two types of ...
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58 views

Show that: if $Cond_2(A)=1$ then $A=\alpha Q$; where $Q $ is an orthonormal matrix and $\alpha$ is a real number $\neq 0$

Show that: if $Cond_2(A)=1$ then $A=\alpha Q$; where $Q$ is an orthonormal matrix and $\alpha$ is a real number$\neq 0$ The condition number is defined as theorem as $Cond_2(A)= ||A|| \hspace{0.1cm}||...
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1answer
111 views

Wilkinson's polynomial very simple misunderstanding

The Wilkinson polynomial $W(x) = \prod_{k=1}^{20}(x-k)$ is notoriously ill conditioned. I wanted to play around with it and see for myself. I wrote a computer code that does bisection method. I gave ...
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46 views

Condition number for the relative residual of the matrix square root

Let $X$ be a square root of $A \in \mathbb{C}^{n \times n}$, i.e., such that $X^2 = A$, and let $$ \alpha(X) = \frac{\|X\|^2}{\|A\|} = \frac{\|X\|^2}{\|X^2\|} \ge 1. $$ On page 135 of Functions of ...

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