Diagonalization of Matrix
Diagonalization of 2×2 Matrix using Similarity (SAS⁻¹ / S⁻¹AS) Diagonalization of a 2×2 Matrix using Similarity Transformation Method: Choose eigenvectors as columns of \(S\), then compute \(D = S^{-1}AS\) (equivalently \(A = SDS^{-1}\)). Given \[ A=\begin{pmatrix}5&4\\[2pt]1&2\end{pmatrix} \] We will diagonalize \(A\) using \(D=S^{-1}AS\), where \(S\) is formed from eigenvectors. Step 1 Find the eigenvalues Characteristic equation: \(\det(A-\lambda I)=0\) \[ \det(A-\lambda I)= \begin{vmatrix} 5-\lambda & 4\\ 1 & 2-\lambda \end{vmatrix} =(5-\lambda)(2-\lambda)-4 \] \[ =(10-7\lambda+\lambda^2)-4=\lambda^2-7\lambda+6 \] ...
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