BETA FUNCTION MATRIX DETERMINANT

21 09 2009

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

\displaystyle a_{i,j}=\frac{1}{\beta(i,j)}, where \beta is the beta function. Then:

|A_{n}|=n!


Proof:

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

A_{n}=U_{n}^{T}*U_n

Where U_n 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:

\displaystyle A_{5}=\left(\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 6 & 12 & 20 & 30 \\ 3 & 12 & 30 & 60 & 105 \\ 4 & 20 & 60 & 140 & 280 \\ 5 & 30 & 105 & 280 & 630 \end{array} \right)= U_{n}^{T}* \left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 0 & \sqrt{2} & 3 \sqrt{2} & 6 \sqrt{2} & 10 \sqrt{2} \\ 0 & 0 & \sqrt{3} & 4 \sqrt{3} & 10 \sqrt{3} \\ 0 & 0 & 0 & 2 & 10 \\ 0 & 0 & 0 & 0 & \sqrt{5} \end{array} \right)

The elements of U_n, seems to be:

\displaystyle u_{i,j}=\sqrt{i}\cdot \binom{j}{i}

U_{n}^T is the transpose of U_n, so:

\displaystyle u_{i,j}^{*}=u_{j,i}=\sqrt{j} \cdot \binom{i}{j}

If we multiply both triangular matrices:

\displaystyle a_{i,j}=\sum_{r=1}^n{u_{i,r}^{*} \cdot u_{r,j}} =\sum_{r=1}^n{r \cdot \binom{i}{r} \binom{j}{r}}=\sum_{r=1}^{min(i,j)}{r \cdot \binom{i}{r} \binom{j}{r}}

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

\displaystyle a_{i,j}=i \cdot \binom{i+j-1}{j-1}=\frac{1}{\beta(i,j)}

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

\displaystyle |A_{n}|=|U_{n}^{T}*U_{n}|=|U_{n}|^{2}=\left(\prod_{i=1}^{n}{u_{i,i}}\right)^{2}=\prod_{i=1}^{n}{u_{i,i}^2}=\prod_{i=1}^{n}{\left(\sqrt{i}\cdot\binom{i}{i}\right)^2}=\prod_{i=1}^{n}{i}=n!


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!


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