Trace(AB) = Trace(BA)?

[computerages]

Hello!

 

I was wondering if someone can help with how Trace(AB) can be equal to Trace(BA)?

 

Thanks!

[Bignose]

Look at the matrix algebra involved with the definition of the trace:

 

[math] tr(\mathbf{A}) = A_{ii} [/math]

 

where the repeated index indicates summation (i.e. Einstein notation)

 

So

 

[math] tr(\mathbf{AB}) = (AB)_{ii} = A_{ij}B_{ji}[/math]

 

now rearrange the order of multiplication

 

[math] = B_{ji}A_{ij} [/math]

 

then, contract on the repeated index i

 

[math] = (BA)_{jj} [/math]

 

finally, note that the j is just a dummy variable (any letter can be used, i, j, k, p, etc.) and that this step looks just like the first step applying the definition of the trace. So

 

[math] (BA)_{jj} = tr(\mathbf{BA}) [/math]

 

Therefore [math] tr(\mathbf{AB}) = tr(\mathbf{BA}) [/math]