Questions: Diagonalize the following matrix, if possible. [ [8 -3 3 2] ]

Diagonalize the following matrix, if possible. [ [8 -3 3 2] ]
Transcript text: Diagonalize the following matrix, if possible. \[ \left[\begin{array}{rr} 8 & -3 \\ 3 & 2 \end{array}\right] \]
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Solution

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Solution Steps

To diagonalize a matrix, we need to find its eigenvalues and eigenvectors. First, calculate the eigenvalues by solving the characteristic equation, which is obtained by setting the determinant of the matrix minus lambda times the identity matrix to zero. Then, for each eigenvalue, find the corresponding eigenvector by solving the system of linear equations. If the matrix is diagonalizable, it will have enough linearly independent eigenvectors to form a basis.

Step 1: Calculate Eigenvalues

The eigenvalues of the matrix \( A = \begin{bmatrix} 8 & -3 \\ 3 & 2 \end{bmatrix} \) are found to be: \[ \lambda_1 = 5 + 4.6781 \times 10^{-8} i, \quad \lambda_2 = 5 - 4.6781 \times 10^{-8} i \]

Step 2: Calculate Eigenvectors

The corresponding eigenvectors for the eigenvalues are: \[ \mathbf{v_1} = \begin{bmatrix} 0.7071 \\ 0.7071 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 0.7071 \\ 0.7071 \end{bmatrix} \] Note that the eigenvectors are complex due to the imaginary parts of the eigenvalues.

Step 3: Determine Diagonalizability

Since the eigenvalues are complex conjugates and the eigenvectors are linearly independent, the matrix \( A \) is diagonalizable.

Final Answer

The matrix \( A \) can be diagonalized, and the eigenvalues and eigenvectors are as follows: \[ \boxed{\text{Eigenvalues: } \lambda_1 = 5 + 4.6781 \times 10^{-8} i, \quad \lambda_2 = 5 - 4.6781 \times 10^{-8} i} \] \[ \boxed{\text{Eigenvectors: } \mathbf{v_1} = \begin{bmatrix} 0.7071 \\ 0.7071 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 0.7071 \\ 0.7071 \end{bmatrix}} \]

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