Given the matrix

$A = \begin{bmatrix} -2 & -5 & -4 \\ 2 & 5 & -1 \\ 0 & 0 & -5 \end{bmatrix}$

we'll proceed with the following tasks:

1) Find the Eigenvalues of Matrix $A$

To find the eigenvalues, we need to solve the characteristic equation:

$\det(A - \lambda I) = 0$

where $\lambda$ represents an eigenvalue and $I$ is the identity matrix. We form the matrix $A - \lambda I$:

$A - \lambda I = \begin{bmatrix} -2 - \lambda & -5 & -4 \\ 2 & 5 - \lambda & -1 \\ 0 & 0 & -5 - \lambda \end{bmatrix}$

The determinant of $A - \lambda I$ is calculated as follows:

$\det(A - \lambda I) = (-2 - \lambda)(5 - \lambda)(-5 - \lambda)$

Since the matrix is lower triangular, the determinant is the product of its diagonal entries:

$\det(A - \lambda I) = (-2 - \lambda)(5 - \lambda)(-5 - \lambda) = 0$

This yields the eigenvalues as the values of $\lambda$ that make each diagonal term zero:

$\lambda_1 = -2, \quad \lambda_2 = 5, \quad \lambda_3 = -5$

Thus, the eigenvalues of $A$ are:

$\lambda = -2, \; 5, \; \text{and} \; -5$

2) Find the Basis of Matrix $A$

To find the basis for $A$, we typically look for a basis for the eigenspaces corresponding to each eigenvalue. This involves finding the null space of $A - \lambda I$ for each eigenvalue.

Basis for Eigenvalue $\lambda = -2$

Set up $A + 2I$:

$A + 2I = \begin{bmatrix} 0 & -5 & -4 \\ 2 & 7 & -1 \\ 0 & 0 & -3 \end{bmatrix}$

We solve $(A + 2I) \mathbf{x} = 0$ to find the eigenvectors for $\lambda = -2$.

Basis for Eigenvalue $\lambda = 5$

Set up $A - 5I$:

$A - 5I = \begin{bmatrix} -7 & -5 & -4 \\ 2 & 0 & -1 \\ 0 & 0 & -10 \end{bmatrix}$

We solve $(A - 5I) \mathbf{x} = 0$ to find the eigenvectors for $\lambda = 5$.

Basis for Eigenvalue $\lambda = -5$

Set up $A + 5I$:

$A + 5I = \begin{bmatrix} 3 & -5 & -4 \\ 2 & 10 & -1 \\ 0 & 0 & 0 \end{bmatrix}$

We solve $(A + 5I) \mathbf{x} = 0$ to find the eigenvectors for $\lambda = -5$.

The collection of these eigenvectors forms a basis for the matrix $A$ based on its eigenspaces.

Would you like me to work out each eigenvector explicitly, or do you have any questions?

---

Follow-up Questions

1. What is the rank of matrix $A$?
2. How would the eigenvalues change if a constant was added to each diagonal element of $A$?
3. What is the geometric multiplicity of each eigenvalue of $A$?
4. How do the eigenvalues relate to the trace and determinant of $A$?
5. Can matrix $A$ be diagonalized based on its eigenvalues?

Tip:

For quick eigenvalue checks, remember that triangular matrices have eigenvalues equal to their diagonal elements.