Inverses and Elementary Matrices
Part 4: Hilbert Matrices (optional)
- A Hilbert matrix is an n x n matrix H
whose (i,j)-th entry is 1/(i+j-1). Enter the definition in the worksheet
to see a typical Hilbert matrix.
- Hilbert matrices are "close" to being singular, even though they
are all invertible. In your worksheet, enter the commands to plot the column vectors of the 3 x 3 Hilbert matrix. Are the column vectors coplanar? What would it mean if they were coplanar?
- Check to see if your computer algebra system
can find the inverse of H. Check the result by multiplying H by the announced
inverse. Is it really the inverse?
- Now convert the exact fractions in your computations
to decimal fractions. Compute H and its inverse again, and check
to see how accurate the inverse is.
modules at math.duke.edu
