Steps to Diagonalize a Matrix
Summary:
- Find the Eigenvalues of A
- Find the Eigenvectors of A
- Construct the Matrix P
- Construct the Diagonal Matrix D
- Verify the Diagonalization
Let us now learn how to Diagonalize a Matrix A step by step.
\[\Large
A =
\begin{bmatrix}
4 & 1 \\
3 & 2
\end{bmatrix}
\]
Step 1: Find the Eigenvalues of A
- For the above Matrix A we can find the Eigenvalues as 5 and 1
- Click here if you need a detailed explanation on finding the Eigenvalues for A
\[\Large
\textcolor{red}{\lambda_1 = 5} \quad \text{and} \quad \textcolor{blue}{\lambda_2 = 1}
\]
Step 2: Find the Eigenvectors of A
\[\Large
\textcolor{red}{\lambda_1 = 5} \quad \text{has eigenvector:} \quad
\textcolor{red}{
x_1 =
\begin{bmatrix}
1 \\
1
\end{bmatrix}
}
\]
\[\Large \textcolor{blue}{\lambda_2 = 1} \quad \text{has eigenvector:} \quad \textcolor{blue}{ x_2 = \begin{bmatrix} 1 \\ -3 \end{bmatrix} } \]
\[\Large \textcolor{blue}{\lambda_2 = 1} \quad \text{has eigenvector:} \quad \textcolor{blue}{ x_2 = \begin{bmatrix} 1 \\ -3 \end{bmatrix} } \]
- Click here if you need a detailed explanation on finding the Eigenvectors for A
Which of the following is the correct first step in diagonalizing a matrix A?