# 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?

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

$$\begin{pmatrix}10^{12}&10^{-12}&\cdots&10^{-12}\\10^{-12}&10^{12}&\ddots&\vdots\\\vdots&\ddots&\ddots&10^{-12}\\10^{-12}&\cdots&10^{-12}&10^{12}\end{pmatrix}$$

($10^{12}$ 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

$$\begin{pmatrix}1&2&\cdots&2\\&1&\ddots&\vdots\\&&\ddots&2\\&&&1\end{pmatrix}$$

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.