Does positive-definite order imply determinant order?

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The discussion centers on the relationship between positive-definite ordering of matrices and their determinants. Specifically, it examines whether the condition A > B for two positive-definite matrices A and B guarantees that det(A) > det(B). Initial findings indicate that this holds true for 2x2 matrices, but further investigation is needed for n x n matrices. The conversation suggests that a sufficient condition for this implication is that all eigenvalues of matrix A must dominate those of matrix B.

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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?

<br /> A&gt;B \Rightarrow \text{det}(A)&gt;\text{det}(B)<br />

Thanks.
 
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Interesting question. I checked for 2x2 matrices and in this case it seems to hold. But for nxn?
 
I guess the a sufficient condition for this is if A&gt;B then do all the eigenvalues of A dominate all of the eigenvalues of B?
 

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