Mathematical marvels¶
Diagonalization of a \(3 \times 3\) Matrix¶
Given a matrix \(A\):
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\]
- Find the eigenvalues by solving the characteristic equation:
\[
\det(A - \lambda I) = 0
\]
- Find the eigenvectors for each eigenvalue \(\lambda\) by solving:
\[
(A - \lambda I)\mathbf{v} = 0
\]
- Form the matrix \(P\) using the eigenvectors as columns:
\[
P = \begin{pmatrix}
\mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3
\end{pmatrix}
\]
- Construct the diagonal matrix \(D\) with the eigenvalues on the diagonal:
\[
D = \begin{pmatrix}
\lambda_1 & 0 & 0 \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3
\end{pmatrix}
\]
- Verify that \(A = PDP^{-1}\).