Exploring near-linearities in price–rate of profit trajectories and the concept of effective rank in input–output matrices

Journal of Economic Structures

The Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)

Table 1 Singular values, Shannon’s entropy and effective rank^a

Ranking of singular values	Singular values (1)	Normalized singular values (2)	Common logarithms of (2) (3)	The product of (2)x(3) (4)
1	1.439	0.477	−0.322	−0.153
2	0.569	0.188	−0.725	−0.137
3	0.287	0.095	−1.022	−0.097
4	0.195	0.064	−1.191	−0.077
5	0.158	0.052	−1.282	−0.067
6	0.110	0.036	−1.439	−0.052
7	0.083	0.027	−1.561	−0.043
8	0.052	0.017	−1.764	−0.030
9	0.040	0.013	−1.873	−0.025
10	0.028	0.009	−2.028	−0.019
11	0.019	0.006	−2.192	−0.014
12	0.017	0.006	−2.237	−0.013
13	0.010	0.003	−2.487	−0.008
14	0.007	0.002	−2.619	−0.006
15	0.004	0.001	−2.846	−0.004
Sum:	3.020	1.000	Shannon (S)	0.746
			erank = e^s	2.107

^aThe effective rank remains the same when employing natural (instead of common) logarithms, given the appropriate adjustment of the base. Consequently, the Shannon entropy S in terms of natural logarithms reformulates relation (12) as\({e}^{S/\mathrm{ln}(10)}\)