Mutual Information Matrices Are Not Always Positive Semidefinite

• Published in: IEEE transactions on information theory Vol. 60; no. 5; pp. 2694 - 2696
• Main Author: Jakobsen, Sune K.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-05-2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computer science; Construction; Educational institutions; Eigenvalues and eigenfunctions; Entropy; Exact sciences and technology; Information theory; Information, signal and communications theory; Linear matrix inequalities; Matrix; Mutual information; Random variables; Telecommunications and information theory; Information inequalities; linear algebra; mutual information; Entropy; Counterexample; Random variable; Mutual information; Linear algebra
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2311434

For discrete random variables X 1 , ..., X n we construct an n by n matrix. In the (i, j)-entry we put the mutual information I(X i ; X j ) between X i and X j . In particular, in the (i, i)-entry we put the entropy H(X i ) = I(X i ; X i ) of X i . This matrix, called the mutual information matrix of (X 1 , ..., X n ), has been conjectured to be positive semidefinite. In this paper, we give counterexamples to the conjecture, and show that the conjecture holds for up to three random variables.