(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant equations

Each of the following tests is a necessary and sufficient condition for the real symmetric matrix A to be positive definite:

a) x^{T}Ax 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.

3. 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!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Positive Definiteness of a Real Matrix

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