Matrices in Modern Economics: a sketch

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• Strange abbreviations are intentional

Least of Linear Algebra

• Sylvester's criterion: real symmetric matrix is positive definite iff all leading principal minors > 0 (corresponds to sequence of variational approximation of multivariable function through its Jacobian matrix)

History (not in chronological order)

• one-sector growth model (no use of matrices): Malthus' EPP
• two-sector growth models (implicit use of linear algebra): Quesnay's TE and Marxian reproduction scheme in DK
• n-sector extensions of reproduction scheme: Sraffa's PCMC and Leontief's IOE
• total differentiability condition of cost and utility functions (Sylvester's criterion): Mathematical Appendix of Hicks' VC
• skew-symmetric (alternative) matrix: payoff matrices of two-player zero-sum games in von Neumann-Morgenstern's TGEB

TODO

• econometric models after Frisch
• modern dynamic models after Samuelson
• explain why no complex values allowed (or, why only "Hermitian operators" operating on "coherent states" allowed in dynamic models)
• other uses?