Shape Operators and Eigenvalues

[chaotixmonjuish]

This is probably falls within a problem of Mathematica as opposed to a question on here but I have a question about the following:

Given some cylinder with the shape operator matrix:

{{0,0},{0,-1/r}}

We get eigenvalues 0 and -1/r and thus eigenvectors {0, -1/r} and {1/r, 0} by my computations.

However we know that the principal directions are {{1},{0}} and {{0},{1}} <= (Computations on Mathematica)

This question, I guess, is an intersection of differential geometry and Linear Algebra.

 

[fzero]

(1,0) and (0,1) are also eigenvectors of that matrix, in fact they are the orthonormal basis for the eigenspace. They are obtained by normalizing the eigenvectors you chose.