be the surface which is the graph ofz = f(x,y) = ax2 + 2bxy + ay2.
Equivalently,f(x,y) = (x,y)A(x,y)t,
is the matrix
- The character of S depends on the eigenvalues of the matrix A:
- If both eigenvalues are positive, S is bowl-shaped, opening up.
- If both eigenvalues are negative, S is bowl-shaped, opening down.
- If one eigenvalue is positive and the other is negative, S issaddle-shaped.
Here is a picture of S
for a particular choice of a
, together with the eigenvalues of the corresponding matrix A
.This picture is a frame from a movie, showing how S
changes as A
varies. Clicking on the image will bring up the movie.
This frame and movie are © 1998 Juha Haataja and the Center for ScientificComputing, Finland.
- Steven Weintraub