A Using projection operators

The mathematical analysis of the dynamics of systems interacting with encompassing reservoirs, whose detailed dynamics is of no direct interest, is facilitated by the use of projection operators or projectors. A simple example is provided by the use of such projectors to rearrange a system of linear equations. Let 

Since we have explicit fonns for all terms on the right, Eq. (9.104) provides an explicit solution for the interesting part, P-part, of our system. 

Problem 9.6. Show that a matrix A can be fomially written in the form 

The mathematical problem was to find the vector x given the vector u. The physical problem may have identified the subspace P as the interesting subspace because the initial information u is given in that space (i.e. Qu = 0) and information about x is needed also in that space (i.e. we require only sPx). 

As an example of an application of this formalism consider again the problem of the decay of an initially prepared level 11) coupled to a continuum /) as in Eqs (9.2)-(9.7) and Fig. 9.1. Let 

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