Timeline for Does positivity of the n(n-1)/2 principal minors formed from 2 x 2 submatrices ensure positive-definiteness of the n x n matrix itself?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2021 at 16:42 | answer | added | Will Jagy | timeline score: 2 | |
| Jul 12, 2021 at 1:56 | comment | added | Sina Baghal | okey, what if you consider 2I+A. Then every 2 by 2 principal is PD. But 2I+A is not even PSD. | |
| Jul 12, 2021 at 1:49 | comment | added | Sina Baghal | for only 2 by 2 principals think about A+I where A is the adjacency of a matrix. Then all the principals of A+I are PSD, but I don't think if A+I will be PSD. For instance for regular bipartite graphs, spectrum of A is symmetric with lambda_min(A)=-k where k is any vertex degree. | |
| Jul 12, 2021 at 1:47 | comment | added | Sina Baghal | i think this is true: if A[1:1], A[1:2,1:2], ... , A[1:n,1:n] are PD, then A is PD. | |
| Jul 12, 2021 at 1:25 | comment | added | Nathaniel Johnston | @PaulB.Slater - Yes. More generally, you can always find $n \times n$ matrices that are not positive definite, but all of their principal minors (other than the full $n \times n$ minor) are positive. | |
| Jul 12, 2021 at 1:23 | comment | added | Paul B. Slater | Thanks, Will Jagy! So, can such counterexamples to positive-definiteness, despite positivity of the principal 2 x 2 minors, be constructed for all n>2? I surmise so. | |
| Jul 12, 2021 at 0:57 | comment | added | Will Jagy | $$ \left( \begin{array}{rrr} 3 & -2 & -2 \\ -2 & 3 & -2\\ -2 & -2 & 3 \end{array} \right) $$ | |
| Jul 12, 2021 at 0:27 | history | asked | Paul B. Slater | CC BY-SA 4.0 |