Summary of chapters 23-27 from Mathematics for Economists by Simon & Blume. Covers all material needed for the final exam of the course Linear Algebra for EOR in the programme Econometrics and Operations Research at the University of Groningen.
Linear Algebra for EOR
(University of Groningen)
Summary 2021-2022
Stuvia: marcellaschrijver
Eigenvalues and eigenvectors ................................................................................................................ 2
Matrices .................................................................................................................................................. 3
General solutions ................................................................................................................................... 5
Phase portraits ....................................................................................................................................... 7
Steady states .......................................................................................................................................... 7
1st order systems of differential equations ........................................................................................... 8
Determinants.......................................................................................................................................... 9
Spaces and subspaces .......................................................................................................................... 11
Eigenvector
Nonzero vector 𝑣 such that (𝐴 − 𝑟𝐼)𝑣 = 0 or 𝐴𝑣 = 𝑟𝑣
Generalized eigenvector
Nonzero vector 𝑤 such that (𝐴 − 𝑟𝐼)𝑤 = 𝑣
○ (𝐴 − 𝑟𝐼)𝑤 ≠ 0 and (𝐴 − 𝑟𝐼)𝑘 𝑤 = 0
Algebraic multiplicity
am(𝑟) = #times the eigenvalue is repeated
Geometric multiplicity
gm(𝑟) = #independent eigenvectors corresponding to the eigenvalue
Quadratic form 𝑸(𝒙) = 𝒙𝑻 𝑨𝒙
○ Positive definite: All eigenvalues > 0
○ Negative definite: All eigenvalues < 0
○ Positive semidefinite: All eigenvalues ≥ 0
○ Negative semidefinite: All eigenvalues ≤ 0
○ Indefinite: Has positive and negative eigenvalues
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