Is it possible to determine if this matrix is ill-conditioned?

I intend to much better recognize ill - conditioning for matrices. Claim we are offered any kind of matrix $A$, where some components are $10^6$ in size and also some are $10^{-7}$ in size. Does this warranty that this matrix has a problem number more than 100? More than 1000? Despite the fact that we have not defined which components of $A$ have those values?

2022-07-25 20:43:18
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I resolved , yet I intend it is excellent to have a solution below. The factor is that neither the dimension of the access neither the dimension of the component is a warranty that your matrix is well - or unwell - conditioned. For that, one would certainly require to consider the matrix is single values, to make use of an usual standard. Specifically, the 2 - standard problem variety of a matrix is the biggest single value separated by the smallest single value ; if the tiniest single value is absolutely no, the matrix is single, and also if the tiniest single value is really little about the biggest single value, you have unwell - conditioning.

As an example, matrices of the kind


($10^{12}$ on the angled, and also $10^{-12}$ off - angled) are well conditioned (the proportion of the biggest to the tiniest single value is really virtually equivalent to $1$), while the family members of upper triangular matrices


researched by and also Jim Wilkinson have a problem number equivalent to $\cot^2\dfrac{\pi}{4n}$, where $n$ is the dimension of the matrix.

2022-07-25 22:32:02