Steps to Diagonalize a Matrix
Step 3: Construct the Matrix P
- P is a matrix formed by combining the eigenvectors of the matrix A.
- Each eigenvector becomes a column in P.
\[\Large
\textcolor{red}{x_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}} \quad
\textcolor{blue}{x_2 = \begin{bmatrix} 1 \\ -3 \end{bmatrix}}
\]
\[\Large \textcolor{purple}{P} = \begin{bmatrix} \textcolor{red}{1} & \textcolor{blue}{1} \\ \textcolor{red}{1} & \textcolor{blue}{-3} \end{bmatrix} \]
\[\Large \textcolor{purple}{P} = \begin{bmatrix} \textcolor{red}{1} & \textcolor{blue}{1} \\ \textcolor{red}{1} & \textcolor{blue}{-3} \end{bmatrix} \]
note: If you have two eigenvalues λ1 = 5 and λ2 = 1, the corresponding eigenvectors x1 and x2 are placed in P in the same order:
- x1 corresponds to λ1 = 5, so it goes in the first column of P.
- x2 corresponds to λ2 = 1, so it goes in the second column of P.
Which of the following statements correctly describes the construction of matrix P during diagonalization?