10. Linear Algebra: Diagonalization of Matrix
Which of the followings is/are correct?
1. For any real matrix A, there exists an invertible real matrix S, so that SAS is a diagonal matrix 2. For any real matrix A, there exists an invertible complex matrix S, so that SAS1 is a diagonal matrix 3. For any real matrix A, there exists an invertible real matrix S, so that SAT AS1 is a diagonal matrix 4. For any symmetric real matrix A, there exists an invertible real matrix S, so that SAS is a diagonal ma
Pick ONE option
All of above
(2), (3) and (4)
(3) and (4)
Only (3)

Let's analyze the statements one by one in the context of matrix diagonalization:

1. **For any real matrix \( A \), there exists an invertible real matrix \( S \), so that \( S^{-1}AS \) is a diagonal matrix.**

   This statement is **false**. A real matrix \( A \) can be diagonalized by a real matrix \( S \) only if \( A \) is diagonalizable and has a complete set of real eigenvalues. Not all real matrices have real eigenvalues or are diagonalizable over the real numbers. Therefore, this statement is not true for any real matrix \( A \).

2. **For any real matrix \( A \), there exists an invertible complex matrix \( S \), so that \( S^{-1}AS \) is a diagonal matrix.**

   This statement is **true**. Any real matrix \( A \) can be diagonalized using a complex matrix \( S \), as every real matrix has a complex Jordan form, and all matrices can be diagonalized over the complex numbers.

3. **For any real matrix \( A \), there exists an invertible real matrix \( S \), so that \( S^{-1}A^T AS \) is a diagonal matrix.**

   This statement is **true**. The expression \( A^T A \) is a symmetric real matrix, and every symmetric real matrix can be diagonalized by an orthogonal matrix (a special kind of invertible real matrix). Thus, there exists an invertible real matrix \( S \) such that \( S^{-1}A^T AS \) is diagonal.

4. **For any symmetric real matrix \( A \), there exists an invertible real matrix \( S \), so that \( S^{-1}AS \) is a diagonal matrix.**

   This statement is **true**. Every symmetric real matrix can be diagonalized by an orthogonal matrix, which is a type of invertible real matrix. Therefore, this statement is correct.

### Conclusion:
The correct statements are **(2)**, **(3)**, and **(4)**.

So, the correct answer is **(2), (3), and (4)**.

Would you like to explore the details further or have any questions? Here are some related questions:

1. What is the Jordan canonical form, and how is it related to diagonalization?
2. How does diagonalization differ between real and complex matrices?
3. What are the necessary and sufficient conditions for a matrix to be diagonalizable?
4. What is the significance of the spectral theorem in diagonalizing symmetric matrices?
5. Can non-square matrices be diagonalized? Why or why not?

**Tip:** Remember that diagonalization is a powerful tool in linear algebra, especially when simplifying matrix operations like powers of matrices.

Explore the concept of matrix diagonalization in linear algebra. Learn about the conditions under which a matrix can be diagonalized, including eigenvalues and orthogonal matrices. This page discusses the theorems related to diagonalization and provides detailed explanations for real and symmetric matrices.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation