Solve Quadratic w/ Grassmann Variables & NxN Matrix

[SeaBeams]

Can anyone advise on the following :

Complete the square for this expression :

(theta)*i Bij (theta)j +(theta)*i(eta)i +(eta)*i(theta)iwhere theta and eta are Grassmann variables and Bij an NxN matrix
with indices i,j=1,...,N.

Many thanks

 

[hamster143]

You can use limited latex in here if you enclose it in tex and /tex tags. Can you try to re-frame your question?

 

[torquil]

Can anyone advise on the following :

Complete the square for this expression :

(theta)*i Bij (theta)j +(theta)*i(eta)i +(eta)*i(theta)i


where theta and eta are Grassmann variables and Bij an NxN matrix
with indices i,j=1,...,N.

Many thanks

In "componentless" notation, your expression is (?):

$$A := \theta^* B \theta + \theta^* \cdot \eta + \eta^* \cdot \theta$$

How about?:

$$A = (\theta^* + \eta^* B^{-1}) B (\theta + B^{-1}\eta) - \eta^*B^{-1}\eta$$

Torquil