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Thread: orthogonal matrix

  1. #1
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    orthogonal matrix

    two problems that I have problem with:

    1. let A be orthogonal matrix that is triangular, prove that A is diagonal.

    2.let $\displaystyle
    R^n = U_1 + U_2

    $ be true, does $\displaystyle
    R^n = U_1 ^ \bot + U_2 ^ \bot

    $ and if it is why???

    Thanks ahead
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  2. #2
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    Quote Originally Posted by omert View Post
    two problems that I have problem with:

    1. let A be orthogonal matrix that is triangular, prove that A is diagonal.
    Say $\displaystyle A$ is upper triangular.

    This means $\displaystyle AA^T = I$.
    The inverse of a upper triangular matrix is upper triangular.
    But $\displaystyle A^T$ is lower triangular.
    Thus, $\displaystyle A^T$ is both upper and lower triangular.
    Thus, $\displaystyle A^T$ is diagnol which means $\displaystyle A$ is diagnol.
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