Decide for or against the positive definiteness of
[2 -1 -1
-1 2 -1 = A
-1 -1 2]
[2 -1 -1
-1 2 1 = B
-1 1 2]
[5 2 1
2 2 2 = C
1 2 5]
Each of the following tests is a necessary and sufficient condition for the real symmetric matrix A to be positive definite:
a) xTAx greater than 0 for all nonzero real vectors x.
b) All the eigenvalues of A are greater than 0
c) All the upper left submatrices of A have positive determinants
d) All the pivots (without row exchanges) are greater than 0.
The Attempt at a Solution
For matrix A,
I found that it fails tests b,c and d. I'm a bit confused because when I performed test a with vector x = [ 1 2 3 ] ^T the test passes, but with an x = [1 1 1]^T the test fails. Therefore, I said that it is not positive definite, but I'm unsure on this one.
For matrices B and C, I said that they are both positive definite because they both pass test c. I'm assuming that if it passes one of the tests it is sufficient.
Am I thinking correctly? Thanks!