Elementary matricies (Ex)
Let be an associative ring with unit and recall that an elementary matrix
over
is a square matrix of the form
![\displaystyle E = \text{Id}_n + a e_{ij}](/images/math/8/6/a/86adb4369aa0836f6d5aa1997fb82470.png)
where is the
identity matrix,
and
is the matrix with zeros in all places except
where it is
and we have
. Clearly each elementary matrix is invertible and so defines an element
where
![\displaystyle GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)](/images/math/4/4/1/441b78c94a7c997bf56a424395d0eb29.png)
is the limit of the invertible matricies.
Exercise 0.1.
Prove that , where
is the subgroup generated by all elements in
which are represented by elementary matrices.
Hint 0.2.
For and
write the matrix
![\displaystyle \left( \begin{array}{cc} ABA^{-1}B^{-1} & 0\\ 0 & I \end{array} \right)](/images/math/c/2/f/c2f7b0f00c0cafd9d1a861ea19a379d9.png)
as a product of elementary matrices
![\displaystyle \left( \begin{array}{cc} I & X\\ 0 & I \end{array} \right) = \prod_{i=1}^{m}\prod_{j=1}^{n} (I+x_{ij}E_{i,j+m})](/images/math/1/0/0/100708562e2d9f067810c477e439ac24.png)
where is an
matrix.
Recall that is defined to be the abelian group
![\displaystyle K_1(R) : = GL(R)_{ab} = GL(R)/[GL(R), GL(R)].](/images/math/a/5/0/a50f6109d26038e574733fb869dbd3a6.png)
Exercise 0.3.
Prove that .
![E](/images/math/e/4/d/e4de1fb0b52291b425e79d56646e3c6e.png)
![R](/images/math/7/d/4/7d47b66519d27f5821540aa071c3d561.png)
![\displaystyle E = \text{Id}_n + a e_{ij}](/images/math/8/6/a/86adb4369aa0836f6d5aa1997fb82470.png)
where is the
identity matrix,
and
is the matrix with zeros in all places except
where it is
and we have
. Clearly each elementary matrix is invertible and so defines an element
where
![\displaystyle GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)](/images/math/4/4/1/441b78c94a7c997bf56a424395d0eb29.png)
is the limit of the invertible matricies.
Exercise 0.1.
Prove that , where
is the subgroup generated by all elements in
which are represented by elementary matrices.
Hint 0.2.
For and
write the matrix
![\displaystyle \left( \begin{array}{cc} ABA^{-1}B^{-1} & 0\\ 0 & I \end{array} \right)](/images/math/c/2/f/c2f7b0f00c0cafd9d1a861ea19a379d9.png)
as a product of elementary matrices
![\displaystyle \left( \begin{array}{cc} I & X\\ 0 & I \end{array} \right) = \prod_{i=1}^{m}\prod_{j=1}^{n} (I+x_{ij}E_{i,j+m})](/images/math/1/0/0/100708562e2d9f067810c477e439ac24.png)
where is an
matrix.
Recall that is defined to be the abelian group
![\displaystyle K_1(R) : = GL(R)_{ab} = GL(R)/[GL(R), GL(R)].](/images/math/a/5/0/a50f6109d26038e574733fb869dbd3a6.png)
Exercise 0.3.
Prove that .