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Euge
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Let ##A, B\in M_n(\mathbb{R})## be positive definite matrices. Prove that for ##0\le t\le 1##, $$\det(tA + (1 - t)B) \ge \det(A)^t\det(B)^{1-t}$$
The log concavity of determinants refers to the property of a matrix where the logarithm of its determinant is a concave function of its entries. In other words, if the entries of a matrix are changed in a specific way, the logarithm of its determinant will always decrease or stay the same.
Log concavity of determinants is useful in various areas of mathematics, such as linear algebra, optimization, and probability. It allows for the simplification and analysis of complex mathematical problems, and also has applications in fields such as economics and statistics.
There are several properties of log concave determinants, including:
A matrix with a log concave determinant is always positive definite, meaning all of its eigenvalues are positive. However, the converse is not always true - a positive definite matrix may not have a log concave determinant. Therefore, log concavity of determinants is a stricter condition than positive definiteness.
Yes, log concavity of determinants can be extended to higher dimensions. In fact, it is a property that holds for matrices of any size, as long as the entries are real numbers. However, the calculations and proofs become more complex as the dimension of the matrix increases.