(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose a 2x2 matrix X (not necessarily Hermitian, nor Unitary) is written as..

[tex]X=a_0+σ \cdot a[/tex]

(In the book σ and a are both bold and are being dotted.)

Where [itex]a_0[/itex] and [itex]a_{1,2,3}[/itex] are numbers.

a.)How are [itex]a_0[/itex] and [itex]a_k, (k=1,2,3)[/itex] related to [itex]tr(X)[/itex] and [itex]tr(σ_kX)[/itex]?

b.)Obtain [itex]a_0[/itex] and [itex]a_k[/itex] in terms of the matrix elements [itex]X_{ij}[/itex].

2. Relevant equations

[itex]tr(X)[/itex]= The trace of X, meaning the sum of its diagonal components.

[itex]tr(X)=\sum_{a'}\left\langle a'|X|a' \right\rangle[/itex]

Where the name a' represents base kets.

3. The attempt at a solution

I do not know where to start to be honest. My first question is how can a 2x2 matrix operator equal a number [itex]a_0[/itex] plus the dot product of two vectors? I know I must be misinterpreting this. Can anyone help?

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# Homework Help: The Math of Quantum Mechanics

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