# Relationship between eigenvectors and singular vectors of a Hermitian matrix?

What is the relationship between the eigenvectors and singular vectors of a Hermitian matrix?

Intuitively, I would expect them to be the same (modulo scaling).
However, this doesn't seem to be the case.

For example, if we compare

>> [U, S, V] = svd([1 2; 2 -1])
U =
-0.447213595499958  -0.894427190999916
-0.894427190999916   0.447213595499958
S =
2.236067977499790                   0
0   2.236067977499789
V =
-1     0
0    -1


with

>> [P, D] = eig([1 2; 2 -1])
P =
0.525731112119133  -0.850650808352040
-0.850650808352040  -0.525731112119133
D =
-2.236067977499790                   0
0   2.236067977499790
inv(P) =
0.525731112119134  -0.850650808352040
-0.850650808352040  -0.525731112119134


we see that the eigenvalues and singular values match (except for a -1 factor), but the vectors don't.

Why is this? Shouldn't they all match?

Change [U, S, V] = svd([1 2; 2 -1]) to [U, S, V] = svd([1 2; 2 -1.0000000000001]) to see this.