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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 every <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 index <math>u \in X.</math>

If <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> are two 1-dimensional matrices over the same index set <math>X,\!</math> then <math>\mathfrak{A} = \mathfrak{B}</math> if and only if <math>\mathfrak{A}_x = \mathfrak{B}_x</math> for every <math>x \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 index <math>x \in X.</math>

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| height="60" | <math>((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_u ~=~ (\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_u</math>

| height="60" | <math>((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~ (\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x</math>

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