If is an square matrix of order , whose elements are defined as:

, where is the beta function. Then:

**Proof:**
If we use Cholesky method, we can decompose this matrix as:

Where is an upper triangular matrix.

But instead of applying the algorithm to a generical case, we are going to propose a factorization, and after we will check that this decomposition generates the appropiate matrix. This mean to use software to speed up the proof, like:

The elements of , seems to be:

is the transpose of , so:

If we multiply both triangular matrices:

This last binomial expression, can be added to a closed form, equal to the reciprocal of beta function, (See proof on reference [1])

Now, that we have found this decomposition, for , then it is unique and is Hermitian and positive definite.

**Archives:**[a]-092109-Beta Determinant.nb

**References:**[1]-Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (Reading, Massachusetts: Addison-Wesley, 1994 – Concrete Mathematics – page 181 Problem 5

[2]-A000142-Factorial numbers: n! The On-Line Encyclopedia of Integer Sequences!

### Like this:

Like Loading...

*Related*