On a given matrix, a, the first way is to take the eigen vectors times the diagonal of the eigen values times the inverse of the original matrix. That should give us back the original matrix. In R that looks like:

a <- matrix( c(1:16), nrow = 4)
p <- eigen(m)$vectors
d <- diag(eigen(a)$values)
p %*% d %*% solve(p)
a

so in that example p %*% d %*% solve(p) should be the same as a