1. ## Orthogonal diagonalization

I can understand the importance of diagonalisation in general (in finding powers of matrices). However why should we orthogonally diagonalise a matrix?
It seems like a lot of work if you have repeated eigenvalues and the corresponding eigenvectors are not orthogonal which belong to this repeated eigenvalue. What use is this?
Why is orthogonal diagonalization so important? Is it applications to quadratic forms?

2. Well, I think the significance lies in the ability to write a matrix in terms of an orthonormal basis of eigenvectors of a symmetric (Hermitian) operator. Simultaneously diagonalizable Hermitian operators have a special significance in quantum mechanics (one of the best applications of linear algebra out there).

That's my two cents.

3. Originally Posted by Euclid
Why is orthogonal diagonalization so important? Is it applications to quadratic forms?

Certainly, that is important. Besides, orthogonal matrices induce isomorphic mappings, which preserve both lengths and angles between vectors (for example rotations, reflections) .

4. Originally Posted by Ackbeet
Well, I think the significance lies in the ability to write a matrix in terms of an orthonormal basis of eigenvectors of a symmetric (Hermitian) operator. Simultaneously diagonalizable Hermitian operators have a special significance in quantum mechanics (one of the best applications of linear algebra out there).

That's my two cents.
Thanks.

5. You're welcome for my contribution.