gptheo3 - Group Theory Handout III Matrix Representations...

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Group Theory Handout III Matrix Representations Consider the set of 3 x 3 matrices below. Construct a multiplication table of these matrices and convince yourself that these matrices form a group. Then compare your multiplication table to the one constructed in class for the two toned triangle 1 0 0 0 1 0 0 1 0 = E 0 0 1 = C 3 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 = C 3 2 0 0 1 = σ 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 = σ 2 1 0 0 = σ 3 1 0 0 0 0 1 Review of Cross products a 11 a 12 a 13 b 11 b 12 b 13 c 11 c 12 c 13 a 21 a 22 a 23 X b 21 b 22 b 23 = c 21 c 22 c 23 a 31 a 32 a 33 b 31 b 32 b 33 c 31 c 32 c 33 Where: c 11 = a 11 b 11 + a 12 b 21 + a 13 b 31 c 12 = a 11 b 12 + a 12 b 22 + a 13 b 32 c 13 = a 11 b 13 + a 12 b 23 + a 13 b 33 c 21 = a 21 b 11 + a 22 b 21 + a 23 b 31 c 22 = a 21 b 12 + a 22 b 22 + a 23 b 32 c 23 = a 21 b 13 + a 22 b 23 + a 23 b 33 c 31 = a 31 b 11 + a 32 b 21 + a 33 b 31 c 32 = a 31 b 12 + a 32 b 22 + a 33 b 32 c 33 = a 31 b 13 + a 32 b 23 + a 33 b 33
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If we carry out matrix multiplication for all of the matrices below we can construct a matrix multiplication table as shown in the next page. E C 3 C 3 2 σ 1 σ 2 σ 3 E E C 3 C 3 2 σ 1 σ 2 σ 3 C 3 C 3 C 3 2 E σ 2 σ 3 σ 1 C 3 2 C 3 2 E C 3 σ 3 σ 1 σ 2 σ 1 σ 1 σ 3 σ 2 E C 3 2 C 3 σ 2 σ 2 σ 1 σ 3 C 3 E C 3 2 σ 3 σ 3 σ 2 σ 1 C 3 2 C 3 E After constructing a matrix multiplication as shown it is a simple matter table to apply the rules of identity, closure, association and inverse and see that the matrices do indeed form a group. We also see that if we substitute the operations as shown then that this table is exactly the same as that observed for the group C 3v we did in class. This set of matrices is said to be a representation of the group C 3v ----- -------- definitions -Any set of elements which multiplied according to the group multiplication table is said to be faithful to the group multiplication table and forms a representation of the group; i. e. the matrices shown on the previous page are a representation of the group C3v. since they are true to the multiplication table for the group C3v. -A matrix is said to be reducible if it can be transformed into a diagonal form.
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gptheo3 - Group Theory Handout III Matrix Representations...

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