Linear Algebra – Finding Eigenvalues and Eigenvectors of A Geometrically

eigenvalues-eigenvectorsgeometrylinear algebramatrices

I am really confused with this question:
Find the eigenvalues and eigenvectors of A geometrically: $$ A = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end {pmatrix} $$^ reflection in the line $y=x$.

Thanks.

Best Answer

I am trying to understand the question. Forgive me if I am totally off. But reflection in $y=x$ means geometrically that any vector on the line, stays where it is, which corresponds with an eigenvalue of 1. Any perpendicular vector to the line, is mapped exactly to its "other" side, which corresponds with an eigenvalue of -1. (The entries of the vector are switched signs)

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