= Solution
The <characteristic polynomial> is $(5-\lambda)(2-\lambda)-4=\lambda^2-7\lambda+6=(\lambda-1)(\lambda-6)$. For $\lambda=1$, the <eigenvector> equation gives $y=2x$; for $\lambda=6$, it gives $x=-2y$. The normalized <eigenvectors> can therefore be chosen as
$$
\boxed{\lambda_1=1,\quad v_1=\frac{(1,2)}{\sqrt5};\qquad\lambda_2=6,\quad v_2=\frac{(-2,1)}{\sqrt5}.}
$$
Both have unit <Euclidean norm>, and their <dot product> is $(-2+2)/5=0$, explicitly verifying <orthogonality>. This also follows from the <spectral theorem> for a real <symmetric matrix> with distinct <eigenvalues>.
Back to article page
