One can think of each row operation as the left product by an elementary matrix.
2.
The elementary matrix for any row operation is obtained by executing the operation on the identity matrix.
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One of the three classes of elementary matrix is involutory, namely the " row-interchange elementary matrix ".
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One of the three classes of elementary matrix is involutory, namely the " row-interchange elementary matrix ".
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Define an " elementary matrix " to be one which is the sum of an identity matrix and a single off-diagonal element ( this is different from the definition used in linear algebra ).
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:To answer my own question, yes it must be possible, since any row operation is equivalent to multiplying an elementary matrix from the left . talk ) 21 : 06, 6 June 2010 ( UTC)
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If " E " is an elementary matrix, as described below, to apply the elementary row operation to a matrix " A ", one multiplies the elementary matrix on the left, " E?" A ".
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If " E " is an elementary matrix, as described below, to apply the elementary row operation to a matrix " A ", one multiplies the elementary matrix on the left, " E?" A ".
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A special case of another class of elementary matrix, that which represents multiplication of a row or column by & minus; 1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.
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I tried looking for these " other contexts " on WP but have only found that it appears to be a case of simple aesthetic preference of the author; square brackets and parentheses seem to be used fairly interchangably in elementary matrix algebra.
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