Results 1 to 7 of 7

Math Help - Properties of Orthogonal Matrix

  1. #1
    Member
    Joined
    Dec 2009
    Posts
    224

    Properties of Orthogonal Matrix



    So, I tried to prove this question in two parts (A) if A is a symmetric matrix then it is orthogonal (B) since A is orthogonal, its row vectors must form an arthonormal set. But I don't think it is possible to prove part (A).

    Anyway here's my proof so far:

    If A is symmetric then A=A^T. A is an orthogonal matrix if AA^T=A^TA=I (since A and A^T are symmetric A and A^T commute). And must have an inverse A^{-1}=A^T.

    Let R_1,R_2,...R_n denote the rows of A; then R^T_1,R^T_2,...R^T_n are the columns of A^T.

    Let AA^T=C_{ij}

    By matrix multipication: C_{ij}=R^T_iR^T_j=R_i.R_j

    Therefore AA^T=I \iff R_i . R_j = \delta_{ij} \iff rows of A form an orthonormal set.

    My proof is proof is probably not correct. Can anyone show me how to prove this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by demode View Post


    So, I tried to prove this question in two parts (A) if A is a symmetric matrix then it is orthogonal (B) since A is orthogonal, its row vectors must form an arthonormal set. But I don't think it is possible to prove part (A).

    Anyway here's my proof so far:

    If A is symmetric then A=A^T. A is an orthogonal matrix if AA^T=A^TA=I (since A and A^T are symmetric A and A^T commute). And must have an inverse A^{-1}=A^T.

    Let R_1,R_2,...R_n denote the rows of A; then R^T_1,R^T_2,...R^T_n are the columns of A^T.

    Let AA^T=C_{ij}

    By matrix multipication: C_{ij}=R^T_iR^T_j=R_i.R_j

    Therefore AA^T=I \iff R_i . R_j = \delta_{ij} \iff rows of A form an orthonormal set.

    My proof is proof is probably not correct. Can anyone show me how to prove this?
    Your proof is correct but the question is wrong. You have correctly proved that if A is an orthogonal matrix then its row vectors form an orthonormal set. But the question asks you to prove the (false) statement that if A is a symmetric matrix then its row vectors form an orthonormal set.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2009
    Posts
    224
    Quote Originally Posted by Opalg View Post
    Your proof is correct but the question is wrong. You have correctly proved that if A is an orthogonal matrix then its row vectors form an orthonormal set. But the question asks you to prove the (false) statement that if A is a symmetric matrix then its row vectors form an orthonormal set.
    How could the question be wrong?? Because this question is given to us for my course at university...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by demode View Post
    How could the question be wrong?? Because this question is given to us for my course at university...
    I hate to disillusion you, but one of life's hard lessons is that nobody, not even a university professor, is infallible; and nothing that you read, not even printed course material, is necessarily 100% correct.

    The matrix \begin{bmatrix}1&1\\1&1\end{bmatrix} is symmetric, but it is not orthogonal and its row vectors do not form an orthonormal set.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Dec 2009
    Posts
    224
    Thanks, I see! Maybe it's some kind of typo, maybe the word "symmetric" was meant to be "orthogonal" or something like that...
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    Quote Originally Posted by Opalg View Post
    I hate to disillusion you, but one of life's hard lessons is that nobody, not even a university professor, is infallible; and nothing that you read, not even printed course material, is necessarily 100% correct.

    The matrix \begin{bmatrix}1&1\\1&1\end{bmatrix} is symmetric, but it is not orthogonal and its row vectors do not form an orthonormal set.
    Hey, you're not supposed to tell people that! Of course, university professors are infallible!!!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by HallsofIvy View Post
    Hey, you're not supposed to tell people that! Of course, university professors are infallible!!!

    Well, many times that's true, but not always. Being an instructor at my alma mater , some time ago already, I had to convince students that not all that's written is infallible, and at long last I told them directly and unmistakenly at the beginning of some course:

    "Listen you all: you must check your notes and books and think and etc....because any mistake that that I, or any teacher, book, or oracle whatsoever may commit when teaching you people stuff is your only and unique responsibility!"

    The above, of course, is a little hard to swallow for freshman and even sometimes for sophomore students, but that way they stopped trying to justify their nonsenses by saying : "the teacher said this, or that book said so...".

    If the teacher makes a mistake it is the serious, commited student's task and duty to find it and mend it...at least by exams time, if not before.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 6th 2010, 08:08 AM
  2. Orthogonal projectors and properties
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 1st 2009, 10:33 AM
  3. orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 6th 2009, 08:28 AM
  4. Orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: April 15th 2008, 06:38 AM
  5. orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 17th 2008, 03:09 AM

Search Tags


/mathhelpforum @mathhelpforum