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Math Help - orthogonal matrices question

  1. #1
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    orthogonal matrices question

    there is an orthogonal matrices nxn,in which all of its members are
    whole numbers (like -1 ,-7,4,8,..).
    also there is vector b b\epsilon R^{n} its members are whole too.

    prove that for the system Ax=b we have a single solution "c"

    where c=(c1,..,cn) and c1 .. cn are whole numbers
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  2. #2
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    Re: orthogonal matrices question

    Quote Originally Posted by transgalactic View Post
    there is an orthogonal matrices nxn,in which all of its members are
    whole numbers (like -1 ,-7,4,8,..).
    also there is vector b b\epsilon R^{n} its members are whole too.

    prove that for the system Ax=b we have a single solution "c"

    where c=(c1,..,cn) and c1 .. cn are whole numbers


    In such a square system and since an orthogonal matrix is always invertible, the solution's given by x=A^{-1}b , and this is an integer

    vector because A^{-1} is an integer matrix (why? What's the determinant of an orthogonal matrix?)

    Tonio
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    Re: orthogonal matrices question

    tonio, i think that invoking the determinant is misleading. orthogonal matrices exist that have non-integral entries, example:

    [-1/√2 -1/√2]
    [ 1/√2 -1/√2].

    more pertinent is the fact that for an orthogonal matrix A, A^-1 = A^T, and clearly A^T has integer entries iff A does.
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    Re: orthogonal matrices question

    Quote Originally Posted by Deveno View Post
    tonio, i think that invoking the determinant is misleading. orthogonal matrices exist that have non-integral entries


    Whoever implied otherwise, and what does this have to do with what I wrote? The important fact for the OP is that the determinant

    of an orthogonal matrix is always \pm 1 , and this means the inverse matrix is also an integer one...

    Tonio





    , example:

    [-1/√2 -1/√2]
    [ 1/√2 -1/√2].

    more pertinent is the fact that for an orthogonal matrix A, A^-1 = A^T, and clearly A^T has integer entries iff A does.
    I don't think the above is more pertinent at all for the supposed level of the OP: if he's asking what he's

    asking, relations between the transpose of a matrix and its inverse and , which usually are studied

    within the frame of vector spaces with inner product, could still be far away from him.

    Tonio
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  5. #5
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    Re: orthogonal matrices question

    sure, i understand, but how do you go from that fact that det(A) = 1, to conclude that A^-1 has integral entries?

    i exhibited an orthogonal matrix for which this is NOT true. in other words, i am wondering what YOU were thinking as the answer

    to the question: "Why (does A^-1 have integer entries)?. What's the determinant of an orthogonal matrix?".

    because if your answer is "because det(A) = 1", that's not correct. having a determinant of 1, does NOT imply integral entries.
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    Re: orthogonal matrices question

    Quote Originally Posted by Deveno View Post
    sure, i understand, but how do you go from that fact that det(A) = 1, to conclude that A^-1 has integral entries?


    By his question he already must know how to calculate the inverse of a matrix: A^{-1}=\frac{1}{\det A}\,Adj(A), so it is now

    obvious that the inverse is going to be an integer matrix (in fact, having determinant \pm 1 is an iff condition for an integer matrix

    to have an integer inverse)...




    i exhibited an orthogonal matrix for which this is NOT true.



    Oh dear! The matrix we've been working with is an INTEGER one: all I've done has taken this into account all through! I never talked

    about general matrices or whatever but focused on the OP where we're given an integer matrix.


    in other words, i am wondering what YOU were thinking as the answer

    to the question: "Why (does A^-1 have integer entries)?. What's the determinant of an orthogonal matrix?".

    because if your answer is "because det(A) = 1", that's not correct. having a determinant of 1, does NOT imply integral entries.

    Of course not! Please do read my above observation in blue to you, and let's hope the OP also

    reads this so that she/he won't confused by your bizarre remarks.

    Tonio
    Last edited by Ackbeet; June 23rd 2011 at 12:38 PM.
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