1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Matrix N is 'M symmetric' ?

  1. Dec 7, 2007 #1
    My dad came across this phrase in a book but neither of us are familiar with it. The statement is :

    "Let [tex]M_{1}[/tex] and [tex]M_{2}[/tex] be matrices. [tex]N = M_{1}^{-1}M_{2}[/tex]. This matrix is [tex]M_{1}[/tex] symmetric and so it diagonalisable in [tex]\mathbb{R}^{2}[/tex]."

    Does it just mean that [tex]M_{1}=M_{1}^{T}[/tex] or something else? Obviously searching for "Matrix, symmetric" doesn't help in this question...
     
  2. jcsd
  3. Dec 7, 2007 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I think more context is needed. Consider you're never actually told what size the matrices are. It says diagonalizable in [tex]\mathbb{R}^{2}[/tex], so I would think they're 2x2, but usually you say diagonalizable over R or over C or over Q, etc. [tex]\mathbb{R}^{2}[/tex] isn't a standard field (and i'm not sure whether it's even possible to make it a field off the top of my head), so my first guess would be that you're missing something important
     
  4. Dec 7, 2007 #3
    I assumed that the matrices are 2x2, so I guess that refers to them being diagonalisable as a Real matrix. Unfortunately I don't have access to this book, he asked me over the phone and what he said differed a few times from what he then emailed me. I assume he's quoting directly, but that might be incorrect too...

    I'll ask him again and see if he's typed out something from memory or he copied it word for word. I agree, there does feel as if there's a vital bit of information missing.
     
  5. Dec 8, 2007 #4

    Chris Hillman

    User Avatar
    Science Advisor

    What book?

    Please make sure you make him tell you what book because I think this is vitally important information!

    My first guess was M-symmetric stands for "Minkowski-symmetric", but if your dad's book has nothing to do with relativistic physics that is fairly unlikely. Another guess is that the (extraneous?) symbol is a printer's error, since (particularly in the context of elementary linear algebra) the sentence with that symbol deleted appears to make sense if all matrices are nxn real matrices and if [itex]N[/itex] is indeed symmetric.

    If it helps, put [itex]L = \operatorname{diag} (-1,1,1, \dots 1)[/itex]; then the Minkowski adjoint can be taken to be [itex]A^{\ast} = L^{-1} \, A^T \, L[/itex] and then a Minkowski symmetric operator satisfies [itex]A^{\ast} = A[/itex]. For example in the 2x2 case a Minkowski-symmetric matrix would take the form
    [tex]
    A = \left[ \begin{array}{cc} a & b \\ -b & d \end{array} \right]
    [/tex]
    while a Minkowski anti-symmetric matrix would take the form
    [tex]
    A = \left[ \begin{array}{cc} 0 & b \\ b & 0 \end{array} \right]
    [/tex]
    which satisfies
    [tex]
    \exp(A) =
    \left[ \begin{array}{cc} \cosh(b) & \sinh(b) \\ \sinh(b) & \cosh(b) \end{array} \right]
    [/tex]
    which can be compared with the analogous facts for the usual transpose.

    I don't think we can offer any useful assistance until your dad supplies the missing context.
     
    Last edited: Dec 8, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Matrix N is 'M symmetric' ?
  1. Non-symmetric matrix (Replies: 1)

Loading...