Let 0<x1<x2<<xn and let A be the symmetric n×n matrix with ijth entry Aij=min{xi,xj}.1 This matrix has linearly independent columns and so is invertible. Its inverse A1 is symmetric, tridiagonal, and has ijth entry [A1]ij={1x1+1x2x1if i=j=11xixi1+1xi+1xiif 1<i=j<n1xnxn1if i=j=n1xjxiif i=j11xixjif i=j+10otherwise. For example, if xi=2i1 for each in=5 then A=[11111122221244412488124816] and A1=[2100011.50.50000.50.750.250000.250.3750.1250000.1250.125] You may wonder: why is this useful? Suppose I observe data {(xi,yi)}i=1n, where the function f:[0,)R mapping regressors xi0 to outcomes yi=f(xi) is the realization of a Wiener process. I use these data to estimate some value f(x) via Bayesian regression. My estimate depends on the inverse of the covariance matrix for the outcome vector y=(y1,y2,,yn). This matrix has ijth entry min{xi,xj}, so I can compute its inverse using the expression above.


  1. Let me know if the family of such matrices has a name! ↩︎