Results 1 to 4 of 4

Math Help - Problem with simple linear algebra exercise

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    34

    Problem with simple linear algebra exercise

    Good day to all. I apologize if the question asked appears to be a waste of time but it has created some problems for me. In one of my end of chapter exercises I am asked the following:

    Find all 2x2 matrices where AA^T (transpose of A) = I

    I proceeded with the general equation of A such that

    A= [[a,b],[c,d]] where [a,b],[c,d] are the rows of A
    A^T = [[a,c],[b,d]]

    After AA^T was computed I retrieved the following three equations:

    (1) a^2 + b^2 = 1
    (2) ac + bd = 0
    (3) c^2 + d^2 =1

    My first question is this a good method to follow in answering the question? My second question is that I am not sure what conclusions to draw from here (aside that AA^T is symmetric). The other problem is that we have not explicitly covered orthogonal matrices or inverses which are topics I encountered when reading other textbooks about this matter. Any help would be greatly appreciated.

    Kindest regards
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2009
    Posts
    277
    Thanks
    2
    Think about eqn 1. This is satisfied by all the points on a circle of radius 1. You can think of (a,b) as a 2 component vector of magnitude 1 pointing to some point on the circle.

    You can reach the same conclusion about (c,d) from eqn.3

    Eqn 2 tells you that (a,b) is perpendicular to (c,d) because the dot product is zero.

    The net effect is that all solutions look like the hour hand and minute hand at 3:00, but with some arbitrary rotation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by gate13 View Post
    Good day to all. I apologize if the question asked appears to be a waste of time but it has created some problems for me. In one of my end of chapter exercises I am asked the following:

    Find all 2x2 matrices where AA^T (transpose of A) = I

    I proceeded with the general equation of A such that

    A= [[a,b],[c,d]] where [a,b],[c,d] are the rows of A
    A^T = [[a,c],[b,d]]

    After AA^T was computed I retrieved the following three equations:

    (1) a^2 + b^2 = 1
    (2) ac + bd = 0
    (3) c^2 + d^2 =1

    My first question is this a good method to follow in answering the question? My second question is that I am not sure what conclusions to draw from here (aside that AA^T is symmetric). The other problem is that we have not explicitly covered orthogonal matrices or inverses which are topics I encountered when reading other textbooks about this matter. Any help would be greatly appreciated.

    Kindest regards

    I think it is an excellent method to answer your question, taking into account that you haven't yet studied orthogonal matrices. Now:

    (2)\,\,ac+bd=0\Longrightarrow a=-\frac{bd}{c}\Longrightarrow\,(1)\,\,1=a^2+b^2=b^2\  left(\frac{d^2}{c^2}+1\right)\Longrightarrow b^2=\frac{c^2}{c^2+d^2} \Longrightarrow b=\pm \frac{c}{\sqrt{c^2+d^2}}\,,\,\,a=\pm \frac{d}{\sqrt{c^2+d^2}} , and a similar expression for b,c using a,b in reversed roles.

    It's clear from (1) and (3) that -1\leq a,b,c,d\leq 1 , and taking a straight-angle triangle with cathetus c,d and hypothenuse \sqrt{c^2+d^2} , we get that we can take

    a=\cos \theta\,,\,\,b=\sin \theta\,,\,\,c=-\sin \theta\,,\,\,d=\cos \theta , which gives you the general form of a 2 x 2 orthogonal real matrix.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2009
    Posts
    34
    Thank you both for your prompt reply to my question. Also, thanks for the geometric interpretation of the problem. Very much appreciated.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple Demonstration Exercise
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: December 1st 2011, 06:57 AM
  2. A simple algebra exercise
    Posted in the Algebra Forum
    Replies: 7
    Last Post: September 17th 2010, 12:41 PM
  3. Simple linear algebra justification - clarification
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 6th 2010, 09:00 PM
  4. Simple linear algebra question - opinion needed
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 3rd 2010, 10:29 AM
  5. Simple Linear Algebra Problem... is the book wrong?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 24th 2009, 01:17 AM

Search Tags


/mathhelpforum @mathhelpforum