For people on this board I have a probably pretty modest question, but since I'm not a mathematician (just an economist), I'm having trouble. The full pdf can be found here: http://www.pnas.org/content/early/2012/05/16/1206569109.full.pdf+html
The question is regarding the following passage and has to do with linear algebra. They write: "where Adj(M′) is the adjugate matrix (also known as the classical adjoint or, as in high-school algebra, the “matrix of minors”). Eq. 2 implies that every row of Adj(M′) is proportional to v. Choosing the fourth row, we see that the components of v are (up to a sign) the determinants of the 3 × 3 matrices formed from the first three columns of M′, leaving out each one of the four rows in turn. These determinants are unchanged if we add the first column of M′ into the second and third columns. The result of these manipulations is a formula for the dot product of an arbitrary four-vector f with the stationary vector v of the Markov matrix, v · f ≡ D(p; q; f), where D is the 4 × 4 determinant shown explicitly in Fig. 2B. This result follows from expanding the determinant by minors on its fourth column and noting that the 3 × 3 determinants multiplying each fi are just the ones described above."
To understand the full context you will probably have to read the beginning of the passage, which is also very short. Yet my question is specifically regarding the formulated relationship between the stationary vector v of the Markov transition-matrix M and the Adj(M'), which is Adj.(M-I). As they say: Every row of Adj.(M') is proportional to v, which is sort of intuitive looking at Eq. 2, but I simply do not understand how they got that. Also the immediately following conclusion that the elements of v are the 3x3 column determinants of M' if you were to eliminate from the bottom of the fourth column.
Also to point out a petty mistake but v · f ≡ D(p; q; f) can't be correct as the dimensions do not link up correctly. v' · f ≡ D(p; q; f) is correct. But yet again I grasp that this formulation makes sense, but fail to understand how this can be arrived at.
If you can point me in the direction of a book or can flat-out explain this to me, I would be very obliged.
Thanks in advance o_s