ABSTRACT
An orthogonal transformation leaves the trace of a square matrix unchanged. If S is the covariance matrix of x, and it is transformed to a diagonal matrix
the trace of S equals the trace of D, since D=U tSU , and
The trace of the covariance matrix S of x=col(x1, …, xn) is the sum of the variances
of the different components xi. It is the same in any orthonormal basis, i.e., whether or not S is diagonal. Therefore, we can define the total variance Stot to
which is independent of the basis vectors used to represent x. In other words, the total variance is invariant under orthogonal transformations. Because of this, it makes sense to define the total variance and to compare the covariance matrices
belonging to the same physical qua tities x1, …, xn when the vectors x are defined in different coördinate systems. The variance of each quantity separately is partly a mathematical property of how we chose to represent these variables, but the total variance depends only on the physics of the situation in question.
