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Revision as of 03:38, 7 May 2009
, 03:38, 7 May 2009→Commentary Note 12.5
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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 of checking whether the 1-dimensional matrices <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W}</math> and <math>\mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math> are equal is to check whether the following equation holds for an arbitrary index <math>x \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 the 1-dimensional matrices <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W}</math> and <math>\mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math> are equal is to check whether the following equation holds for an arbitrary index <math>x \in X.</math>
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~ (\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x</math>
| height="60" | <math>((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~ (\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x</math>
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Taking both ends toward the middle, we proceed as follows:
{| align="center" cellspacing="6" width="90%"
| height="80" |
<math>
(\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~
\prod_{p \in X} (\mathfrak{S}^\mathfrak{L})_{xp}^{\mathfrak{W}_p} ~=~
\prod_{p \in X} (\prod_{q \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp}})^{\mathfrak{W}_p} ~=~
\prod_{p \in X} \prod_{q \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp}\mathfrak{W}_p}
</math>
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{| align="center" cellspacing="6" width="90%"
| height="80" |
<math>
(\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x ~=~
\prod_{q \in X} \mathfrak{S}_{xq}^{(\mathfrak{L}\mathfrak{W})_q} ~=~
\prod_{q \in X} \mathfrak{S}_{xq}^{\sum_{p \in X} \mathfrak{L}_{qp} \mathfrak{W}_p} ~=~
\prod_{q \in X} \prod_{p \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp} \mathfrak{W}_p}
</math>
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