Proving Isomorphisms

  • #1

Homework Statement



Hello, I have been asked to prove that three different matrices which are skew-symmetric with a defined operation can be shown to be isomorphic to the usual vectors in 3d space with the operation of the cross product.

Homework Equations



Well the operation i guess is not so important to state as I have constructed a multiplication table for it but it is X*Y = XY-YX.

The Attempt at a Solution



I know that the cross product for a x b would be a2b3 - a3b2 etc etc and i can see in my head why it is isomorphic I just really do not know how to go about proving it. what should be vectors be, should they all be the same at just i,j,k? i dont know how to prove it without using numbers as the vectors.
please help
 

Answers and Replies

  • #2


also how can i write a skew-symmetric matrix as a linear combination of I J and K?
please help thanks
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
966


If, by "I, J, and K" you mean the 3-d basis vectors, you can't- matrices are not vectors. But you can write such a matrix as a linear combination of three basis matrices that the isomorphism maps into [itex]\vec{i}[/itex], [itex]\vec{j}[/itex], and [itex]\vec{k}[/itex].

Any 3 by 3 skew-symmetric matrix is of the form
[tex]\begin{bmatrix}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{bmatrix}[/tex]
[tex]= a\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0\end{bmatrix}+ c\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}[/tex]
 
  • #4


thanks so muchhh i think i know what im doing now!! cant believe never knew that way of writing skew-symmetric matrices
 
  • #5
Deveno
Science Advisor
908
6


actually, that operation IS important, it is the typical way of defining a Lie Bracket on a matrix algebra:

[X,Y] = XY - YX

by identifying R^3 with a certain subset of the matrix algebra, you have shown that the cross product can be used to identify R^3 as a subalgebra of the Lie Algebra M3(R).
 

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