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Math Help - Orthogonality proof

  1. #1
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    Orthogonality proof

    i have a linera algebra quiz on this proof tomorrow so if anyone could show me how the following is done, i would be really thankful...

    An orthogonal matrix is one for which A^T = A^-1 (meaning A transpose = inverse of A).

    Prove that if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.

    Is this true for the converse of this statement as well? Justify your answer.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by buckaroobill View Post
    i have a linera algebra quiz on this proof tomorrow so if anyone could show me how the following is done, i would be really thankful...

    An orthogonal matrix is one for which A^T = A^-1 (meaning A transpose = inverse of A).

    Prove that if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.
    Let A be orthogonal. and consider:

    (A^t A)_{i,j}

    this is the dot product of the i-th col and j-th col of A, hence as A^t=A^{-1}:

    (A^t A)_{i,j} = 1, if i = j

    (A^t A)_{i,j} = 0, if i != j

    Is this true for the converse of this statement as well? Justify your answer.
    The converse is not true, Consider any orthogonal matrix A, then B=2A has the
    given property but:

    B^t B = 4 I.

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Let A be orthogonal. and consider:

    (A^t A)_{i,j}

    this is the dot product of the i-th col and j-th col of A, hence as A^t=A^{-1}:

    (A^t A)_{i,j} = 1, if i = j

    (A^t A)_{i,j} = 0, if i != j



    The converse is not true, Consider any orthogonal matrix A, then B=2A has the
    given property but:

    B^t B = 4 I.

    RonL
    Thanks! What do the underscores and the exclamation point preceding the equal sign mean, though?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by buckaroobill View Post
    Thanks! What do the underscores and the exclamation point preceding the equal sign mean, though?
    A_{i,j} means A subscript i,j, it's LaTeX but our LaTeX is not working at present, but it is also a common ASCII math convention.

    != not equal to.

    RonL
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