Let's analyze the question and each option:

Question:

Which of the following statements about the diagonalization of a matrix is true?

Option Analysis:

1. A: All matrices are diagonalizable.

   • This is false. Not all matrices are diagonalizable. For example, a matrix with a defective eigenvalue (one where the geometric multiplicity is less than the algebraic multiplicity) cannot be diagonalized.

2. B: A matrix of size $n$ is diagonalizable if it has $n$ distinct eigenvalues.

   • This is true. If a matrix has $n$ distinct eigenvalues, it guarantees that it has enough linearly independent eigenvectors to form a basis for diagonalization.

3. C: Diagonalization is only possible for symmetric matrices.

   • This is false. While symmetric matrices over the real numbers are always diagonalizable, diagonalization is not limited to symmetric matrices. For example, certain non-symmetric matrices with distinct eigenvalues are also diagonalizable.

4. D: A matrix can be diagonalized if it has only real eigenvalues.

   • This is false. Having real eigenvalues does not necessarily mean the matrix is diagonalizable. For example, if a matrix has a defective eigenvalue, it won't be diagonalizable even if all eigenvalues are real.

Correct Answer:

B: A matrix of size $n$ is diagonalizable if it has $n$ distinct eigenvalues.

Would you like further clarification or details on diagonalization concepts?

Related Questions:

1. What is the relationship between eigenvalues and eigenvectors in diagonalization?
2. Can you provide an example of a matrix that is not diagonalizable?
3. How does the geometric multiplicity of eigenvalues affect diagonalizability?
4. Are all symmetric matrices over complex numbers diagonalizable?
5. What are the steps to determine if a given matrix is diagonalizable?

Tip:

Always check if a matrix has enough linearly independent eigenvectors to confirm diagonalizability. This can be verified by ensuring the number of eigenvectors matches the matrix size.