# Tag Info

Accepted

### Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

If the eigenvalues of $A$ are $\lambda_1, \lambda_2, \dots,\lambda_n$, the eigenvalues of $A + \mu I$ are $\lambda_1 + \mu, \lambda_2 + \mu, \dots, \lambda_n + \mu$. It is an easy computation to ...
• 11.4k

### Why are systems with clustered eigenvalues easy to solve?

A good explanation of this phenomena with many examples is given in Iterative Methods for Linear and Nonlinear Equations by Tim Kelley. The crux of it comes down to the fact that each step of a ...
• 2,443
Accepted

### How to directly compute the inverse of an ill-conditioned dense matrix

Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally. I would use the term badly-conditioned instead of ill-...
• 8,672

### Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

The accepted answer is right: you are not making a preconditioner. To elaborate. For a matrix $A$, a preconditioner is a matrix $B$ such that $B^{-1}A$ has a smaller condition number than $A$. The ...
• 1,330
Accepted

### Poorly conditioned, easily evaluated sum for unit testing

The condition number of sum $s(x) = \sum_{j=1}^n x_j$ is given by $$\kappa(x) = \frac{\sum_{j=1}^n |x_j|}{|\sum_{j=1}^n x_j|} = \frac{s(|x|)}{|s(x)|}$$ and reflects the sums sensitivity to small ...
• 1,391
Accepted

### Why can ill-conditioned linear systems be solved precisely?

Added after my initial answer: It appears to me now that the author of the referenced paper is giving condition numbers (apparently 2-norm condition numbers but possibly infinity-norm condition ...
• 18.7k