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Math Help - Orthogonal diagonalization

  1. #1
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    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?
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  2. #2
    A Plied Mathematician
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    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.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Euclid View Post
    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) .
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  4. #4
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    Quote Originally Posted by Ackbeet View Post
    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.
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  5. #5
    A Plied Mathematician
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    You're welcome for my contribution.
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