F=ℝ: Normal matrix with real eigenvalues but not diagonalizable

In summary, the conversation discusses whether a normal matrix with real eigenvalues is necessarily diagonalizable. The participants discuss the difficulty in finding a counterexample and mention the possibility of using the complex field instead of the real field. A hint is also given to look for a proof instead of a counterexample.
  • #1
Bipolarity
776
2
I am going through Friedberg and came up with a rather difficult problem I can't seem to resolve.
If ## F = ℝ ## and A is a normal matrix with real eigenvalues, then does it follow that A is diagonalizable? If not, can I find a counterexample?

I'm trying to find a counterexample, by constructing a matrix A that is normal and has real eigenvalues, but is not diagonalizable. It is giving me some problems! Any ideas?

If I did not have the assumption that A has real eigenvalues, the rotation matrix would suffice as a counterexample.

If I had the complex field instead of the real field, then easily A is diagonalizable since normality implies orthogonal diagonalizibility in a complex inner product space.

BiP
 
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  • #2
Just a hint: there is no counterexample, look for proof.
 

1. What is the significance of a normal matrix with real eigenvalues but not diagonalizable?

A normal matrix with real eigenvalues but not diagonalizable is an important concept in linear algebra and matrix theory. It represents a special type of matrix that has real eigenvalues (which have many real-world applications) but cannot be diagonalized. This means that it cannot be written in the form of a diagonal matrix with the eigenvalues on the main diagonal. This type of matrix has unique properties and can be used in various applications, such as in quantum mechanics and signal processing.

2. What does it mean for a matrix to be normal?

A matrix is considered normal if it commutes with its conjugate transpose. In other words, if A is a normal matrix, then A*A = A*A, where A* is the conjugate transpose of A. This property is important because it allows for the matrix to be diagonalized and for its eigenvalues to be easily calculated. Normal matrices also have other unique properties, such as having orthogonal eigenvectors.

3. How can you determine if a matrix has real eigenvalues?

A matrix has real eigenvalues if all of its eigenvalues are real numbers. This can be determined by finding the characteristic polynomial of the matrix and solving for its roots. If all of the roots are real numbers, then the matrix has real eigenvalues. Another way to determine this is by checking if the matrix is Hermitian, which means it is equal to its own conjugate transpose. Hermitian matrices always have real eigenvalues.

4. Why is it important for a normal matrix to have real eigenvalues?

The real eigenvalues of a normal matrix have many important applications in mathematics, science, and engineering. For example, in quantum mechanics, the eigenvalues of a Hermitian matrix (which is a type of normal matrix) represent the possible energy states of a quantum system. In signal processing, the real eigenvalues of a normal matrix can be used to analyze signals and filter out noise. Additionally, the real eigenvalues of a normal matrix can help determine the stability and behavior of systems in control theory.

5. Can a normal matrix with real eigenvalues always be diagonalized?

No, not all normal matrices with real eigenvalues can be diagonalized. Some normal matrices have repeated eigenvalues or complex eigenvalues, which cannot be diagonalized. However, all normal matrices with distinct real eigenvalues can be diagonalized. This is known as the spectral theorem for normal matrices. Additionally, normal matrices with repeated eigenvalues can often be transformed into a similar diagonal matrix using a Jordan decomposition.

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