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 $s$ .

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+'''Whitehead's lemma''' states that a [[matrix (mathematics)|matrix]] of the form <math>
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+\begin{bmatrix}
+u & 0 \\
+& u^{-1} \end{bmatrix}</math> is equivalent to [[identity matrix|identity]] by elementary transformations:
+:<math>
+\begin{bmatrix}
+u & 0 \\
+& u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}). </math>
+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>.
+{{algebra-stub}}
+[[Category:Linear algebra]]