The matrix \( X^{-1}AX \) is:

\[
X^{-1}AX = \begin{pmatrix} 5 & 1.05 \times 10^{-15} \\ 3.58 \times 10^{-16} & 1 \end{pmatrix}
\]

This is effectively a diagonal matrix with the eigenvalues \( 5 \) and \( 1 \) on the diagonal. The very small off-diagonal elements are due to numerical approximations and can be considered zero.

This confirms that the matrix \( X \) is correct and that \( X^{-1}AX \) is diagonal as required.

Would you like more details, or do you have any other questions?

Here are some related questions you might be interested in:

1. How is the inverse of a matrix calculated?
2. What is the significance of eigenvalues and eigenvectors in linear algebra?
3. How do numerical errors arise in matrix computations?
4. What are the properties of diagonal matrices?
5. How is diagonalization used in solving differential equations?
6. What are some applications of eigenvalues in physics?
7. Can all matrices be diagonalized?
8. What is the relationship between diagonalization and matrix similarity?

**Tip:** Understanding the process of diagonalization is crucial for simplifying many problems in linear algebra, especially those involving powers of matrices.

Explore the concept of matrix diagonalization and eigenvalues with a detailed example. Learn how to verify diagonalization, interpret eigenvalues, and handle numerical approximations in matrix computations. Ideal for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation