True, but with a condition.

The statement is generally true for symmetric matrices (or Hermitian matrices in the case of complex vector spaces). For symmetric matrices, the Spectral Theorem guarantees that they are diagonalizable and that their eigenvectors can be chosen to be orthogonal. This means that if you have a symmetric matrix, you can always find an orthogonal set of eigenvectors corresponding to the matrix’s eigenvalues.

However, for non-symmetric matrices, even though they may be diagonalizable, the eigenvectors cannot always be made orthogonal. In such cases, the eigenvectors corresponding to distinct eigenvalues may still be linearly independent, but they do not necessarily have to be orthogonal.

Summary:

• For symmetric matrices, eigenvectors can always be made orthogonal.
• For non-symmetric diagonalizable matrices, eigenvectors cannot always be orthogonal.

Would you like to dive deeper into how this works with examples or specific cases? Let me know!

Here are some related questions:

1. What is the Spectral Theorem, and how does it relate to symmetric matrices?
2. How can you check if a matrix is diagonalizable?
3. What is the difference between orthogonal and orthonormal eigenvectors?
4. How do you find eigenvectors of a diagonalizable matrix?
5. What conditions ensure that eigenvectors are linearly independent?

Tip: Symmetric matrices always have real eigenvalues, which makes finding orthogonal eigenvectors easier!