Does positive-definite order imply determinant order?

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In summary, positive-definite order refers to a property of a symmetric matrix where all of its eigenvalues are positive, while determinant order refers to the property of a matrix where the determinant is positive. Positive-definite order implies determinant order and a matrix cannot have positive-definite order without also having determinant order. These properties have various applications in fields such as statistics, economics, and physics.
  • #1
hadron23
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Hi,

Given two real-valued positive-definite matrices A and B, assume one is greater than the other with respect to positive definite ordering. That is, A>B. Does the following implication hold?

[tex]
A>B \Rightarrow \text{det}(A)>\text{det}(B)
[/tex]

Thanks.
 
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  • #2
Interesting question. I checked for 2x2 matrices and in this case it seems to hold. But for nxn?
 
  • #3
I guess the a sufficient condition for this is if [tex]A>B[/tex] then do all the eigenvalues of [tex]A[/tex] dominate all of the eigenvalues of [tex]B[/tex]?
 

1. What is the definition of positive-definite order?

Positive-definite order refers to a property of a symmetric matrix where all of its eigenvalues are positive. This means that for any non-zero vector, the resulting product of the matrix and the vector will always be positive.

2. What is determinant order?

Determinant order refers to the property of a matrix where the determinant of the matrix is positive. This means that the matrix is invertible and has a non-zero determinant.

3. Does positive-definite order imply determinant order?

Yes, positive-definite order implies determinant order. This is because a matrix with positive eigenvalues will always have a positive determinant, making it invertible and non-singular.

4. Can a matrix have positive-definite order but not determinant order?

No, a matrix cannot have positive-definite order without also having determinant order. This is because the determinant is directly related to the eigenvalues of a matrix.

5. What are the applications of positive-definite and determinant order?

Positive-definite and determinant order have various applications in fields such as statistics, economics, and physics. In statistics, these properties are used to define positive-definite matrices in multivariate analysis. In economics, they are used in game theory and linear programming. In physics, they are used in quantum mechanics and the study of energy states in materials.

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