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For the reflection about the line $y=-x$, the vector $(1,-1)$ lies along the reflection line and remains unchanged. Therefore, $(1,-1)$ is an eigenvector with an eigenvalue of $1$.

Conditions: The transformation is a reflection about the line $y=-x$.; The vector is nonzero.; The vector lies on the axis of reflection.

For the reflection about the line $y=-x$, the vector $(1,-1)$ lies along the reflection line and remains unchanged. Therefore, $(1,-1)$ is an eigenvector with an eigenvalue of $1$.

Conditions: The transformation is a reflection about the line $y=-x$.; The vector is nonzero.; The vector lies on the axis of reflection.