Name of the matrix:

Gaussian matrix

Pattern:

 

$$\displaystyle a_{ij}=a_{i-j}=\sqrt{\frac{\pi}{2\sigma}} e^{-\frac{\sigma}{2}(i-j)^2}, \qquad i,j=0,1,\ldots$$

Properties:

symmetric, positive definite, the asymptotic condition number is

$${\mathrm{cond}}(A) = \left(\prod_{j=1}^\infty \frac{1+{\mathrm{e}}^{-(j-\frac{1}{2})\sigma}}{1-{\mathrm{e}}^{-(j-\frac{1}{2})\sigma}} \right)^2$$

Source/utilization:

test for the solution of linear systems

References:

 
C.V.M. van der Mee, Z.M. Nashed and S. Seatzu, Sampling expansions in unitarily translation invariant reproducing kernel Hilbert spaces, Adv. Comput. Math. 19(4), 355-372 (2003).
C.V.M. van der Mee and S. Seatzu, A method for generating infinite positive self-adjoint test matrices and Riesz bases, SIAM J. Matr. Anal. Appl., to appear.

Matlab:

t=tptest(1,n,sigma)