The question provides a matrix $A$ and asks about the eigenspace $E_k$ of $A$. The matrix $A$ is given as:

$A = \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix}.$

Analysis

1. The matrix $A$ is a diagonal matrix with eigenvalue $k$ repeated three times.

2. The eigenspace $E_k$ corresponds to all vectors that satisfy $(A - kI)x = 0$. Substituting $A = kI$, the equation becomes:

   $(kI - kI)x = 0 \implies 0x = 0.$

   Thus, the eigenspace $E_k$ is the entire $\mathbb{R}^3$, i.e., all possible linear combinations of the standard basis vectors $\{e_1, e_2, e_3\}$.

Options

The correct answer must include all three basis vectors $e_1, e_2, e_3$, as the eigenspace $E_k$ spans the entire space.

The correct answer is E: $\{e_1, e_2, e_3\}$ is a basis for the eigenspace $E_k$.

Let me know if you want a step-by-step explanation or have any follow-up questions!

Relative Questions

1. What are the properties of eigenvalues and eigenvectors for diagonal matrices?
2. How is the eigenspace of a matrix related to its eigenvalues?
3. Can eigenvalues of a matrix be repeated? What is the geometric multiplicity of such eigenvalues?
4. What is the difference between the eigenspace and the null space of a matrix?
5. How would the eigenspace change if the matrix had distinct eigenvalues?

Tip

Always check the geometric multiplicity (dimension of the eigenspace) and ensure it matches the algebraic multiplicity (repeated occurrences of the eigenvalue).