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> small matrices that cannot be inverted with 32-bit floats, or even 64-bit floats Presumably these all have eigenvalues near zero and away from zero (i.e. the
by jey 6y ago
> small matrices that cannot be inverted with 32-bit floats, or even 64-bit floats
Presumably these all have eigenvalues near zero and away from zero (i.e. the condition number is large)?
- the_svd_doctor 6y agoYes. Ill conditioned matrices are hard to “invert” (more like solve Ax=b for a given A and b) almost by definition. The condition number is basically a measure if sensitivity. If I change b a little, the change in x is proportional to b multiplied by the condition number. If the matrix is very ill conditioned (say more than 1e10 for a double precision algorithm), the matrix is really singular for any practical purpose, so the problem is really ill posed.
- msspecter 6y agohttps://en.wikipedia.org/wiki/Hilbert_matrix https://en.wikipedia.org/wiki/Hilbert_matrix "The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation."