  This paper gives a personal assessment of Epoch making advances in Matrix
Computations from antiquity and with an eye towards tomorrow.
  We trace the development of number systems and elementary algebra, and the
uses of Gaussian Elimination methods from around 2000 BC on to current
real-time Neural Network computations to solve time-varying linear equations.
  We include relevant advances from China from the 3rd century AD on, and from
India and Persia in the 9th century and discuss the conceptual genesis of
vectors and matrices in central Europe and Japan in the 14th through 17th
centuries AD.
  Followed by the 150 year cul-de-sac of polynomial root finder research for
matrix eigenvalues, as well as the superbly useful matrix iterative methods and
Francis' eigenvalue Algorithm from last century.
  Then we explain the recent use of initial value problem solvers to master
time-varying linear and nonlinear matrix equations via Neural Networks.
  We end with a short outlook upon new hardware schemes with multilevel
processors that go beyond the 0-1 base 2 framework which all of our past and
current electronic computers have been using.
