Then the eigenvectors of f consist precisely of the entries on the diagonal of that upper-triangular matrix
I think this is a typo and should be “eigenvalues” instead of “eigenvectors”?
The determinant is negative when the operator flips all the vectors it works on.
This could be misleading. E.g. the operator f(v) := -v that literally just flips all vectors has determinant (-1)^n, where n is the dimension of the space it’s working on. The sign of the determinant tells you whether an operator flips the orientation of volumes, it can’t tell you anything about what it does to individual vectors.
(Regarding “orientation of volumes”: in the 2D case, think of R^2 as a sheet of paper, then f(v) := -v is just a 180 degree rotation, so the same side stays up, and the determinant is positive. In contrast, flipping along an axis requires turning over the paper, so negative determinant. Unfortunately this can’t really be visualized the same way in 3D, so then you have to think about ordered bases.)
I think this is a typo and should be “eigenvalues” instead of “eigenvectors”?
This could be misleading. E.g. the operator f(v) := -v that literally just flips all vectors has determinant (-1)^n, where n is the dimension of the space it’s working on. The sign of the determinant tells you whether an operator flips the orientation of volumes, it can’t tell you anything about what it does to individual vectors.
(Regarding “orientation of volumes”: in the 2D case, think of R^2 as a sheet of paper, then f(v) := -v is just a 180 degree rotation, so the same side stays up, and the determinant is positive. In contrast, flipping along an axis requires turning over the paper, so negative determinant. Unfortunately this can’t really be visualized the same way in 3D, so then you have to think about ordered bases.)
Thanks—right on both counts! Post amended.