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Positive Definiteness of a Real Matrix

  • #1

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) 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!

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
I like Serena
Homework Helper
6,577
176
Hi tatianaiistb! :smile:

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.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

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) 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!

Homework Statement





Homework Equations





The Attempt at a Solution

Note: (a) is the definition of positive-definiteness; it is not a test at all.
 
  • #4
So, if it fails one test it is sufficient to say that it is not positive definite, and viceversa? Thanks!!!!
 
  • #5
I like Serena
Homework Helper
6,577
176
Yep!
 

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