# Multiple integral

Let denote the usual inner product of . Evaluate the integral

where is a positive symmetric matrix and .

Solution

Since is a positive symmetric matrix , so is . For a positive symmetric matrix there exists an positive symmetric matrix such that . Applying this to our integral becomes

where is the Euclidean norm. Applying a change of variables we have that

Since then by converting to polar coordinates we have that

Here denotes the surface area measure of the unit sphere and it is known to be

hence

where denotes the Gamma Euler function for which it holds that