Let's assume that all principal minors of symetric square matrix $A$ ($n\times n$) are equal to zero, then what definiteness does this matrix have? It's obvious that it's semidefinite, of course. But which exactly semidefiniteness is it? Negative, positive, or maybe something like "neutral"???!
Asked
Active
Viewed 2,221 times
1
-
Please show some of your thoughts on the question. – Vishal Gupta Sep 08 '13 at 13:11
-
Are all principal minors zero, or are only all leading principal minors zero? The answers are different. – user1551 Sep 08 '13 at 14:07
-
@Vishal I have no idea. All my thoughts were said in the questions... – Viktor Svientsitskyi Sep 08 '13 at 17:25
-
@user1551 Yes, all principal minors. In symetric matrix all principal minors are equal, hence we can use only all leading principal minors for the definiteness investigation. – Viktor Svientsitskyi Sep 08 '13 at 17:28
1 Answers
2
Hint. If all principal minors are zero, show that the matrix is zero and hence it is both positive semidefinite and negative semidefinite.
If only the leading principal minors are guaranteed to be zero, the matrix can be positive semidefinite, negative semidefinite (it is easy to construct a diagonal matrix example) or indefinite (for the matrix to be indefinite, it is at least $3\times 3$).
user1551
- 139,064
-
Ok, thank you, I thought that ¨both¨ variant is immposible, but now I know that it an be. – Viktor Svientsitskyi Sep 08 '13 at 18:35