In mathematics, the Heinz mean (named after E. Heinz) of two non-negative real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s A and B, was defined by Bhatia as:


with 0 ≤ x ≤ 1/2.

For different values of x, this Heinz mean interpolates between the arithmetic

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 (x = 0) and geometric

Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

 (x = 1/2) means such that for 0 < x < 1/2:


The Heinz mean may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.