Alg1131W12T3 - Matrix commutation identity

Sunday, 27 May 2018

10:05 PM

One of the big differences between matrix arithmetic and ordinary arithmetic is that with matrices, multiplication is no longer commutative; in general. Thus if —1 then AB 1 2 —3 and BA 5 -3 and B O Showing that indeed Note: the Maple notation for the matrix 112 >,< 314 And what about associativityQ This does hold for addition of matrices; but what about multiplication? If then —6 In addition BC 4 O —1 20) and so A(BC) 9 AB doesn't always equal SA ' O This supports our understanding that @ yes, associativity of matrix multiplication does hold O no: associativity of matrix multiplication does not hold in general.

 

Consider the 2 x 2 matrices 1 1 o They satisfy the properties that . 12 and 1 o —1 1 Do these relations look familiar? They sure do: they are exactly the same relations satisfied by O the real numbers 1 and O @ the complex numbers 1 and i O the real numbers 1 and —1 O the complex numbers 0 and i It follows that the 2 x 2 matrices contain the arithmetic of complex numbers' So for example the complex number —9 + 3i would correspond to the matrix 1 Note: the Maple notation for the matrix 3 2 112 >,< 314 4

 

Consider the 2 x 2 matrices 1 1 o They satisfy the properties that and 1 o 2 1 . 12 . K2 . 1K 21 Do these relations look familiar? They sure do: they are exactly the same relations satisfied by O the complex numbers 1 and 2i O the real numbers 1 and O @ the real numbers 1 and O the complex numbers 0 and i It follows that the 2 x 2 matrices contain the arithmetic of certain kinds of irrational numbers' So for example the number —4 + 60 would correspond to the matrix 1 Note: the Maple notation for the matrix 3 2 112 >,< 314 4

 

a Suppose that A c b is a 2 x 2 d 1 matrix with the property that AX we can deduce that we can conclude (again) that XA for any 2 x 2 matrix X We say that A commutes with any other matrix X O as well as Looking at the special case X o Looking at another special case X o Putting these ftvo facts together, we deduce that A must have a rather special form. Give a non-zero example of such a matrix A, whose elements are chosen from the set {0, 3, 7} 1 Note: the maple notation for the matrix 3 2 112 >,< 314 4

 

In quantum mechanics, 2 x 2 matrices play a big role. In particular the theory of electron spin uses the Pauli matrices where i is the usual complex number satisfying i Then we can check that: Note: the Maple notation for the complex matrix 3i << 112*1 >,< We can also observe that -iZ -iX • xz 4i

 

 

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