Questions: The matrix A=[2 0 0 0; -4 2 0 2; 6 -2 0 -2; 6 -2 -2 0] has two distinct eigenvalues λ1<λ2. Find the eigenvalues and a basis for each eigenspace. λ1= , whose eigenspace has a basis of λ2= , whose eigenspace has a basis of

The matrix A=[2 0 0 0; -4 2 0 2; 6 -2 0 -2; 6 -2 -2 0] has two distinct eigenvalues λ1<λ2. Find the eigenvalues and a basis for each eigenspace. λ1= , whose eigenspace has a basis of λ2= , whose eigenspace has a basis of
Transcript text: The matrix $A=\left[\begin{array}{cccc}2 & 0 & 0 & 0 \\ -4 & 2 & 0 & 2 \\ 6 & -2 & 0 & -2 \\ 6 & -2 & -2 & 0\end{array}\right]$ has two distinct eigenvalues $\lambda_{1}<\lambda_{2}$. Find the eigenvalues and a basis for each eigenspace. $\lambda_{1}=$ $\square$ , whose eigenspace has a basis of $\lambda_{2}=$ $\square$ , whose eigenspace has a basis of $\square$ $\square$ . Submit answer Next item
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Solution

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Solution Steps

To find the eigenvalues of matrix \( A \), we need to solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \). Once we have the eigenvalues, we can find the corresponding eigenvectors by solving \( (A - \lambda I)x = 0 \) for each eigenvalue.

Solution Approach
  1. Compute the characteristic polynomial of \( A \).
  2. Solve the characteristic polynomial to find the eigenvalues.
  3. For each eigenvalue, solve the system \( (A - \lambda I)x = 0 \) to find the eigenvectors.
Step 1: Compute the Eigenvalues

The eigenvalues of the matrix \( A \) are calculated from the characteristic polynomial \( \text{det}(A - \lambda I) = 0 \). The computed eigenvalues are: \[ \lambda_1 = -5.5991 \times 10^{-16} - 1.7632 \times 10^{-8}j \] \[ \lambda_2 = -5.5991 \times 10^{-16} + 1.7632 \times 10^{-8}j \]

Step 2: Determine the Eigenspaces

The corresponding eigenvectors (eigenspaces) for each eigenvalue are found by solving \( (A - \lambda I)x = 0 \). The eigenspaces are given by: For \( \lambda_1 \): \[ \text{Eigenspace for } \lambda_1 = \begin{bmatrix} 0 \\ -0.5774 + 5.0900 \times 10^{-9}j \\ 0.5774 - 0.0000j \\ 0.5774 + 3.4103 \times 10^{-24} \end{bmatrix} \] For \( \lambda_2 \): \[ \text{Eigenspace for } \lambda_2 = \begin{bmatrix} 0 \\ -0.5774 - 5.0900 \times 10^{-9}j \\ 0.5774 + 0.0000j \\ 0.5774 - 3.4103 \times 10^{-24} \end{bmatrix} \]

Final Answer

The eigenvalues and their corresponding eigenspaces are: \[ \lambda_1 = -5.5991 \times 10^{-16} - 1.7632 \times 10^{-8}j, \quad \text{Eigenspace for } \lambda_1 = \begin{bmatrix} 0 \\ -0.5774 + 5.0900 \times 10^{-9}j \\ 0.5774 - 0.0000j \\ 0.5774 + 3.4103 \times 10^{-24} \end{bmatrix} \] \[ \lambda_2 = -5.5991 \times 10^{-16} + 1.7632 \times 10^{-8}j, \quad \text{Eigenspace for } \lambda_2 = \begin{bmatrix} 0 \\ -0.5774 - 5.0900 \times 10^{-9}j \\ 0.5774 + 0.0000j \\ 0.5774 - 3.4103 \times 10^{-24} \end{bmatrix} \] Thus, the final boxed answer is: \[ \boxed{\lambda_1 = -5.5991 \times 10^{-16} - 1.7632 \times 10^{-8}j, \quad \lambda_2 = -5.5991 \times 10^{-16} + 1.7632 \times 10^{-8}j} \]

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