# 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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### 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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### 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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### 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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### 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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### 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 ...