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
, Eigenvalues and eigenvectors
Eigenvalue
Value 𝑟 such that 𝑑𝑒𝑡(𝐴 − 𝑟𝐼) = 0
𝑘 × 𝑘 matrix 𝐴 with eigenvalues 𝑟1 , … , 𝑟𝑘
○ Trace: 𝑟1 + 𝑟2 + ⋯ + 𝑟𝑘
○ Determinant: 𝑟1 ∙ 𝑟2 ∙ ⋯ ∙ 𝑟𝑘
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
(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
, Eigenvalues and eigenvectors
Eigenvalue
Value 𝑟 such that 𝑑𝑒𝑡(𝐴 − 𝑟𝐼) = 0
𝑘 × 𝑘 matrix 𝐴 with eigenvalues 𝑟1 , … , 𝑟𝑘
○ Trace: 𝑟1 + 𝑟2 + ⋯ + 𝑟𝑘
○ Determinant: 𝑟1 ∙ 𝑟2 ∙ ⋯ ∙ 𝑟𝑘
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
