Determinants
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.
- Enter the commands in your worksheet to plot the column vectors of the 3 x 3 Hilbert matrix. Examine the vectors. Are they coplanar? What would it mean if they were coplanar? How is the geometry of the column vectors related to the determinant of the matrix?
- Compute the determinant of H. Explain how your result suggests that H is "close" to being singular. Is H actually singular?
- Have your computer algebra system compute the inverse of H. Check the result by multiplying H by the announced 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.
- Compute the determinant of your converted Hilbert matrix. Does the result differ from the computation in Step 3?
modules at math.duke.edu
