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For the reflection about $y=-x$, the vector $(1,1)$ is mapped to $(-1,-1)$. Since $(-1,-1) = -1 \cdot (1,1)$, the vector $(1,1)$ 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 is perpendicular to the reflection line.

For the reflection about $y=-x$, the vector $(1,1)$ is mapped to $(-1,-1)$. Since $(-1,-1) = -1 \cdot (1,1)$, the vector $(1,1)$ 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 is perpendicular to the reflection line.