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So, I was looking at the paper by Andrzej CICHOCKI and in the preliminary and notation part where they define some identities, the following identity is also given:

$$\mathrm {tr}(P)-\log \det(P) \geq n$$ where $P$ may be any positive definite matrix.

I was wondering how one would prove the above identity. I have read about the identity in the book Graphical Models by Joe Whittaker, but there it is only written that the identity is a corollary of a special case of Kullback Leibler divergence. So, how can we prove the above identity using the divergence?

user1551
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Frank wiene
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1 Answers1

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I am not aware of Kullback Leibler divergence. But here is a proof of the inequality: $$ \frac{x_1 + ... + x_n}{n} \geq \left( x_1 + ... + x_n \right)^{1/n} $$ or $$ \frac{tr(P)}{n} \geq \det(P)^{1/n} $$ (using the expression of the trace and determinant in terms of eigenvalues) $$ \log \left( \frac{tr(P)}{n} \right) \geq \frac{1}{n} \log( \det(P) ) $$ Next use $$ x - 1 \geq \log( x ), \quad \, \forall x \geq 0, $$ $$ \frac{1}{n} tr(P) - 1 \geq \frac{1}{n} \log( \det(P) ) \quad \Rightarrow tr(P) - n \geq \log( \det(P) ) $$

user7440
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