Timeline for Checking singularity of a matrix
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2020 at 17:17 | comment | added | Federico Poloni | @TommiHöynälänmaa Then that's precisely the distance to singularity! (In Euclidean or Frobenius norm). | |
| Aug 6, 2020 at 16:07 | comment | added | tohoyn | It is the smallest singular value. Sorry for the error. | |
| Aug 6, 2020 at 14:27 | comment | added | Federico Poloni | @TommiHöynälänmaa Smallest eigenvalue of $A$, or of $[0 A; A^T 0]$ (i.e. smallest singular value of $A$)? If it's the latter, then the matrix is at distance exactly $\sigma_{\min}$ from the nearest singular matrix. If it's the former, then it is impossible to tell; there could be a singular matrix at arbitrarily small distance from it. That's the reason why singular values are better than eigenvalues in assessing singularity. :) | |
| Aug 6, 2020 at 11:32 | comment | added | tohoyn | I get the smallest absolute eigenvalue $\sigma_{\mathrm{min}} \approx 0.03$. Is the matrix close to be singular? | |
| Aug 6, 2020 at 6:34 | comment | added | Federico Poloni | @TommiHöynälänmaa That seems correct to me. | |
| Aug 6, 2020 at 5:38 | comment | added | tohoyn | I need to do the computations in C++ and the matrices are not given explicitly. I'm doing ARPACK iteration for vector $(x_n^1,...,x_n^N,y_n^1,...,y_n^N)$ with $$\mathbf{x}_{n+1} = A \mathbf{y}_n$$ and $$\mathbf{y}_{n+1} = A^T \mathbf{x}_n$$ I set the sigma parameter to "SM" in order to compute the smallest (absolute value) singular values. Is this correct? | |
| Aug 5, 2020 at 11:44 | comment | added | Federico Poloni | Sort of. The smallest eigenvalue does not provide a meaningful metric of how close a matrix is to singularity, unlike the smallest singular value. | |
| Aug 5, 2020 at 11:09 | comment | added | tohoyn | I solved the problem by computing the eigenvalue with smallest magnitude with ARPACK. Is this correct? | |
| Aug 5, 2020 at 10:52 | comment | added | Mark L. Stone | I have now provided an expanded version of my comment as an answer. | |
| Aug 5, 2020 at 2:50 | comment | added | Mark L. Stone | Yes, but I was giving an alternative to IEEE arithmetic. The OP did not state that only IEEE arithmetic was allowed. | |
| Aug 4, 2020 at 19:11 | comment | added | Federico Poloni | @MarkL.Stone I consider "interval arithmetic" to be something distinct from "IEEE arithmetic", even if it is implemented using it. | |
| Aug 4, 2020 at 17:52 | comment | added | Mark L. Stone | Interval or radial arithmetic with outward rounding ,might be able to definitively determine a matrix iis not singular. But I don't believe it would be able to definitivlely determine it is singular. I.e., two possible results: 1) not singular 2) may or may not be sngular. See for example ti3.tu-harburg.de/paper/rump/Ru10a_alt.pdf using INTLAB under MATLAB. for verified computation of singular values. | |
| Aug 4, 2020 at 17:33 | history | edited | Federico Poloni | CC BY-SA 4.0 |
added 260 characters in body
|
| Aug 4, 2020 at 15:46 | history | answered | Federico Poloni | CC BY-SA 4.0 |