Prove that this matrix is positive defined

Submitted by TommyR on Thu, 08/23/2007 - 21:14.

Algebra questions

Prove that this matrix is positive defined:

$ A= \begin{pmatrix} \:1 &-\frac{1}{2}& -\frac{1}{2}& \:1 &-\frac{1}{2} &-\frac{1}{2} \\ -\frac{1}{2} & \:1 &-\frac{1}{2} &-\frac{1}{2} & \:1 &-\frac{1}{2} \\ -\frac{1}{2} &-\frac{1}{2} & \:1 &-\frac{1}{2} &-\frac{1}{2} & \:1 \\ \:1 &-\frac{1}{2} &-\frac{1}{2}& \:1 &-\frac{1}{2}& -\frac{1}{2} \\ -\frac{1}{2} & \:1 &-\frac{1}{2} &-\frac{1}{2} & \:1 & -\frac{1}{2} \\ -\frac{1}{2} &-\frac{1}{2} & \:1 &-\frac{1}{2} &-\frac{1}{2} & \:1 \end{pmatrix} $

A matrix A is positive defined if:
$<Ax,x>\geq0, \forall x=(x_1, x_2, x_3, x_4, x_5, x_6)^{T}$

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Prove that this matrix is positive defined