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, 18:08, 6 May 2009→Commentary Note 12.5
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If <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> are two square matrices over the same index set <math>X,\!</math> then <math>\mathfrak{A} = \mathfrak{B}</math> if and only if <math>\mathfrak{A}_{uv} = \mathfrak{B}_{uv}</math> for all <math>u, v \in X.</math> Therefore, a routine way to check whether <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W} = \mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math> is to check whether the equation holds for an arbitrary pair of indices <math>u, v \in X.</math>
If <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> are two square matrices over the same index set <math>X,\!</math> then <math>\mathfrak{A} = \mathfrak{B}</math> if and only if <math>\mathfrak{A}_{uv} = \mathfrak{B}_{uv}</math> for all <math>u, v \in X.</math> Therefore, a routine way to check whether <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W} = \mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math> is to check whether the following equation holds for an arbitrary pair of indices <math>u, v \in X.</math>
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