SPD Matrix Geometric Mean
Generalize a scalar geometric mean to positive matrices that need not commute.
Free theory articleGeneralize a scalar geometric mean to positive matrices that need not commute.
Concept & interpretation
The geometric mean of two positive definite matrices generalizes the scalar concept √(ab). When matrices commute, it is exactly the product of their square roots. When they do not commute, non-trivial rotations must be accommodated.
Experiment
Explore this using the interactive module:
- Open the SPD Matrix Geometric Mean module.
- Set Pair to "Reciprocal" and Angle to 0. Notice the arithmetic mean trace is 5, but geometric mean trace is 4.
- Change the Angle to 45 degrees. The geometric mean changes because the matrices no longer commute!
- The visualization will show how the geometric mean ellipse always lies "between" the ellipses for A and B, but in a geodesic sense rather than a linear Euclidean sense.