Editing Determinant (section)

=== Upper and lower bounds ===
For a positive definite matrix {{math|''A''}}, the trace operator gives the following tight lower and upper bounds on the log determinant
:<math>\operatorname{tr}\left(I - A^{-1}\right) \le \log\det(A) \le \operatorname{tr}(A - I)</math>

with equality if and only if {{math|1=''A'' = ''I''}}. This relationship can be derived via the formula for the [[Kullback-Leibler divergence]] between two [[multivariate normal]] distributions.

Also,
:<math>\frac{n}{\operatorname{tr}\left(A^{-1}\right)} \leq \det(A)^\frac{1}{n} \leq \frac{1}{n}\operatorname{tr}(A) \leq \sqrt{\frac{1}{n}\operatorname{tr}\left(A^2\right)}.</math>

These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the [[harmonic mean]] is less than the [[geometric mean]], which is less than the [[arithmetic mean]], which is, in turn, less than the [[root mean square]].