Find a basis for the eigenspace corresponding to the eigenvalue a+bi

To find a basis for the eigenspace corresponding to the complex eigenvalue \(\lambda = 8 + 8i\), we follow these steps:

1. **Form the matrix** \(\mathbf{A} - \lambda \mathbf{I}\), where \(\mathbf{A}\) is the given matrix and \(\lambda = 8 + 8i\).
2. **Solve the equation** \((\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0}\) to find the eigenvectors corresponding to \(\lambda\).

Given:
\[
\mathbf{A} = \begin{pmatrix}
0 & 1 \\
-128 & 16
\end{pmatrix}
\]
\[
\lambda = 8 + 8i
\]

**Matrix \(\mathbf{A} - \lambda \mathbf{I}\)**:
\[
\mathbf{A} - \lambda \mathbf{I} = \begin{pmatrix}
0 - (8 + 8i) & 1 \\
-128 & 16 - (8 + 8i)
\end{pmatrix}
= \begin{pmatrix}
-8 - 8i & 1 \\
-128 & 8 - 8i
\end{pmatrix}
\]

Let's solve \((\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0}\) for the vector \(\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\).The basis for the eigenspace corresponding to the eigenvalue \(8 + 8i\) is given by the vector:

\[
\begin{pmatrix}
\frac{1}{16} - \frac{i}{16} \\
1
\end{pmatrix}
\]

This vector forms a basis for the eigenspace since it is the only independent vector found from the null space calculation.

Would you like to explore how to find the basis for the eigenspace corresponding to the other eigenvalue, \(8 - 8i\)? If you have any other questions, feel free to ask!

Here are some related questions you might consider:

1. What is the interpretation of an eigenspace in the context of linear transformations?
2. How do you verify if a vector is indeed an eigenvector corresponding to a given eigenvalue?
3. Can eigenspaces corresponding to different eigenvalues intersect?
4. How do you compute the geometric multiplicity of an eigenvalue?
5. What is the algebraic multiplicity of an eigenvalue, and how does it relate to the geometric multiplicity?
6. What role do eigenvectors and eigenvalues play in solving systems of differential equations?
7. How do you normalize an eigenvector?
8. What is the significance of complex conjugate pairs of eigenvalues in dynamical systems?

**Tip:** When dealing with complex eigenvalues, the eigenvectors can also be complex. These eigenvectors and their conjugates often correspond to rotations and scaling in the complex plane.

Learn how to find a basis for the eigenspace corresponding to the complex eigenvalue λ = 8 + 8i in linear algebra. Detailed step-by-step solution provided, including matrix formation, solving the eigenvalue equation, and interpretation of eigenspaces. Suitable for advanced undergraduate students and learners of linear algebra.

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