[R] A problem with chol() function

[Ron Michael]

I think I am missing something with the chol() function. Here is my calculation:
 
> mat
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    3    0    0    0
[2,]    0    1    0    0    0
[3,]    0    0    1    0    0
[4,]    0    0    0    1    0
[5,]    0    0    0    0    1
> eigen(mat)
$values
[1] 1 1 1 1 1
$vectors
     [,1]          [,2] [,3] [,4] [,5]
[1,]    1 -1.000000e+00    0    0    0
[2,]    0  7.401487e-17    0    0    0
[3,]    0  0.000000e+00    1    0    0
[4,]    0  0.000000e+00    0    1    0
[5,]    0  0.000000e+00    0    0    1
> chol(mat)
Error in chol.default(mat) : 
  the leading minor of order 2 is not positive definite

As per the eigen values my matrix is PD (as all eigen values are positive). Then why still I can not get Cholesky factor of my matrix? Can somebody point mw where I am missing?
 
Thanks and regards,