A First Course in Linear Algebra · Problem CB.C41

Representations — Robert Beezer

Mathematics — Linear Algebra · eigenvectors-basis-symmetric-matrices

Problem

Let

S_{22}

be the vector space of

2 \times 2

symmetric matrices. Find a basis for

S_{22}

composed of eigenvectors of the linear transformation

Q: S_{22} \to S_{22}

Q\left(\begin{bmatrix}a&b\\b&c\end{bmatrix}\right) = \begin{bmatrix}25a+18b+30c & -16a-11b-20c\\ -16a-11b-20c & -11a-9b-12c\end{bmatrix}