= Solution
For a <matrix> with equal diagonal entries, the symmetric and antisymmetric directions $(1,1)$ and $(1,-1)$ are <eigenvectors>. Direct multiplication of $AA^{\mathsf T}$ gives <eigenvalues> $3/2-1/2=1$ and $3/2+1/2=2$. Dividing each direction by its <Euclidean norm> gives
$$
\boxed{\lambda_1=1,\quad \mathbf u_1=\frac1{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix};\qquad
\lambda_2=2,\quad \mathbf u_2=\frac1{\sqrt2}\begin{pmatrix}1\\-1\end{pmatrix}.}
$$
The two normalized <eigenvectors> are <orthogonal>; either may independently be replaced by its negative.
Back to article page
