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In this paper, we introduce properly-invariant diagonality measures ofHermitian positive-definite matrices. These diagonality measures are defined asdistances or divergences between a given positive-definite matrix and itsdiagonal part. We then give closed-form expressions of these diagonalitymeasures and discuss their invariance properties. The diagonality measure basedon the log-determinant $\alpha$-divergence is general enough as it includes adiagonality criterion used by the signal processing community as a specialcase. These diagonality measures are then used to formulate minimizationproblems for finding the approximate joint diagonalizer of a given set ofHermitian positive-definite matrices. Numerical computations based on amodified Newton method are presented and commented.

Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem