Positive Definiteness of a Real Matrix

[tatianaiistb]

Homework Statement

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]

Homework Equations

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

 

[I like Serena]

Hi tatianaiistb!

Yes, it is sufficient if a matrix passes one of the tests.
Each test is equivalent to each other test.

Note that for test (a) the test has to pass for ALL nonzero real vectors.
In other words, this is not a practical test.

 

[Ray Vickson]

Homework Statement

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]

Homework Equations

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

Note: (a) is the definition of positive-definiteness; it is not a test at all.

 

[tatianaiistb]

So, if it fails one test it is sufficient to say that it is not positive definite, and viceversa? Thanks!

 

[I like Serena]

Yep!