Let be the geometric mean of two positive semidefinite matrices and . The authors extend the definition of to any number of positive semidefinite matrices inductively. Suppose that for some , the geometric mean of any positive semidefinite matrices has been defined. Let be a -tuple of positive semidefinite matrices. Define .
The authors show that the sequence has a limit of the form and define . The definition given here is the only one in the literature that has the properties that one would expect from a geometric mean. The authors also prove some new properties of the geometric mean of two matrices, and give some simple computational formulae related to them for matrices.