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Whitehead's lemma: Difference between revisions

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'''Whitehead Lemma''' states that a matrix of the form <math>
'''Whitehead's lemma''' states that a [[matrix (mathematics)|matrix]] of the form <math>
\begin{bmatrix}
\begin{bmatrix}
u & 0 \\
u & 0 \\
0 & u^{-1} \end{bmatrix}</math> is equivalent to Identity by elementary transformations.
0 & u^{-1} \end{bmatrix}</math> is equivalent to [[identity matrix|identity]] by elementary transformations:


<math>
:<math>
\begin{bmatrix}
\begin{bmatrix}
u & 0 \\
u & 0 \\
0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}) </math>
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 1 and <math>ij^{th}</math> entry is <math>s</math>.
Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is 1 and <math>ij^{th}</math> entry is <math>s</math>.


{{math-stub}}
{{algebra-stub}}
[[Category:Linear algebra]]

Revision as of 20:28, 7 July 2006

Whitehead's lemma states that a matrix of the form is equivalent to identity by elementary transformations:

Here, indicates a matrix whose diagonal block is 1 and entry is .