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[parent] example of a matrix commutator (Example)

Calculate the commutator [A,B] of the matrices

⌊ ⌋ ⌊ ⌋ 5 3i − i 1 0 1 A = ⌈ 1 0 2 ⌉ B = ⌈ − 2 2i 3 ⌉ i 2i − 1 3i 1 − 1

Ans.

The commutator is calculated by definition as

[A, B] = AB − BA

Carrying out the first matrix multiplication gives

⌊ 8 − 6i − 6 − i 5 + 10i ⌋ ⌈ ⌉ AB = 1 + 6i 2 − 1 − 6i − 5 1 + 7i

and the second multiplication is

⌊ ⌋ 5 + i 5i − 1 − i BA = ⌈ − 10 + 5i 0 − 3 + 6i ⌉ . 1 + 14i − 9 − 2i 6

Finally, subtracting the two yields

⌊ 3 − 7i − 6 − 6i 6 + 11i ⌋ ⌈ ⌉ [A, B] = AB − BA 11 + i 2 2 − 6i . − 1 − 20i 4 + 2i − 5 + 7i

Since the commutator is non-zero, we say that A and B do not commute.


"example of a matrix commutator" is owned by bloftin.
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Cross-references: commute, matrix multiplication, matrices, commutator

This is version 1 of example of a matrix commutator, born on 2010-02-14.
Object id is 838, canonical name is ExampleOfAMatrixCommutator.
Accessed 1978 times total.

Classification:
Physics Classification: 03.65.Ca (Formalism)
 03.65.Fd (Algebraic methods )
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