Whitehead's lemma: Difference between revisions

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Whitehead's lemma states that a matrix of the form ${\begin{bmatrix}u&0\\0&u^{-1}\end{bmatrix}}$ is equivalent to identity by elementary transformations:

{\begin{bmatrix}u&0\\0&u^{-1}\end{bmatrix}}=e_{21}(u^{-1})e_{12}(1-u)e_{21}(-1)e_{12}(1-u^{-1}).

Here, $e_{ij}(s)$ indicates a matrix whose diagonal block is $1$ and $ij^{th}$ entry is $s$ .

It also refers to the closely related result that the derived group of the general linear group is the group generated by elementary matrices. In symbols, $\operatorname {E} (A)=[\operatorname {GL} (A),\operatorname {GL} (A)]$ . This holds for the stable group (the direct limit of matrices of finite size) over any ring.

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 \begin{bmatrix}
 u & 0 \\
-& u^{-1} \end{bmatrix}</math> is equivalent to [[identity matrix|identity]] by elementary transformations:
+& u^{-1} \end{bmatrix}</math> is equivalent to [[identity matrix|identity]] by [[elementary matrices|elementary transformations]]:
 :<math>
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 Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>.
+It also refers to the closely related result that the [[derived group]] of the [[general linear group]] is the group generated by [[elementary matrices]]. In symbols, <math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>.
+This holds for the stable group (the [[direct limit]] of matrices of finite size) over any ring.
 {{algebra-stub}}