How could one define a dense symmetric positive definite matrix (dimension $1000 \times 1000$) with uniformly distributed eigenvalues (with the smallest eigenvalue $1$ and the condition number $100$) in Matlab?
Because the matrix has to be symmetric, it's absolute eigenvalues are also the singular values. So $\sigma_1=100$ and $\sigma_{1000}=1$. The rest of the eigenvalues/singular values are just random numbers between 1 and 100. The last step is to check whether the matrix is positive definite, which is correct because all of it's eigenvalues are positive.
Any help on the algorithm and the implementation/code would be much appreciated!