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A physical quantity can be represented by the following form:

$A = a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3$ where $\sigma$ matrices are Pauli matrices. Also suppose that there is $B = b_1\Sigma_1 + b_2\Sigma_2 + b_3\Sigma_3$.

A. I read that $a_1,a_2,a_3$ are direction vectors, and when the sum of A's representing real quantity and B's is zero, as each quantity can only be 1 or -1, probability $AB$ being 1 or -1 depend on the value of $-a_1b_1 - a_2b_2 - a_3b_3$.

The question would be: isn't $a_1,a_2.., b_1,b_2,..$ only direction vectors? If so, how can probability of finding $AB$ being 1 change? I do get this mathematically, but I am unable to picture the situation with $a,b$ being direction vectors.

B. I read that when the sum of A's representing real quantity and B's is zero, determining a value on one direction of $A$ set the value of the whole $A$. Is this true? Can anyone explain this?

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The question is somewhat unclear. Where does A+B=0 come from? This just means B=-A, so all the eigenvalues have the opposite sign. The eigenvalues of $\sigma_i$ are all 1 or -1. What do you mean by quantities? If you consider $A=a_1 \sigma_1=9001 \sigma_1$, then the eigenvalues are 9001 and -9001. Also the composite (selfadjoint) matrices can be diagonalized. Btw. the product AB is given by this relation, again linear in $\sigma_i$, see wikipedia. – Nick Kidman Sep 28 '12 at 9:58
@NickKidman modified the question. Hope it's clear. – mother Sep 28 '12 at 10:38
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No, it's still not clear. Also, why do you write "A's"? There is only one A. The sigmas, as well as this A or the B are 2x2 matrices. If you sum matrices you get another matrix. A priori, this doesn't relate to the eigenvalues, may they be 1,-1 or 9001. You should point out if your argumentation has to do with these matrixes only, or with some eigenvalues, or with some expecatation values. To what extend is this a mathematical question and to what extend do you mix in physics? And what physics? – Nick Kidman Sep 28 '12 at 11:41
If $\Sigma_i=\sigma_i$ for all $i=1,2,3$, then $AB$ can be written as a linear combination of identity matrix and Pauli matrices. See this wiki page for derivation. Also unless direction cosines $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are perpendicular to each other eigenvalues of $AB$ will be different from $1,-1$. – user10001 Sep 28 '12 at 17:49

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