This document is a summary of all lecture slides provided by Stefan Pichler during the 2021/2022 course Linear Algebra. The course is part of the first year of the Econometrics and Operations Research program at the Rijksuniversiteit Groningen.
Linear Algebra Summary
Econometrics and Operations Research 2021/2022
,Table of Contents
Week 1..................................................................................................................................................... 3
Eigenvectors and Eigenvalues ............................................................................................................. 3
Diagonalizable Matrices ...................................................................................................................... 3
Difference Equations ........................................................................................................................... 3
Nondiagonalizable Matrices ................................................................................................................ 4
Jordan Normal Forms .......................................................................................................................... 4
Complex Eigenvalues and Eigenvectors .............................................................................................. 5
Week 2..................................................................................................................................................... 6
Markov Processes................................................................................................................................ 6
Symmetric Matrices ............................................................................................................................ 6
Quadratic Forms .................................................................................................................................. 7
Differential Equations.......................................................................................................................... 7
Solutions of First Order ODEs .............................................................................................................. 8
Solutions of Inhomogeneous Linear ODEs .......................................................................................... 8
Week 3..................................................................................................................................................... 9
Second Order Linear ODEs .................................................................................................................. 9
Direction Fields .................................................................................................................................. 10
Phase Portraits .................................................................................................................................. 10
Systems of Differential Equations ..................................................................................................... 11
Linear Systems of ODEs ..................................................................................................................... 12
Week 4................................................................................................................................................... 13
Stability Properties of Equilibrium Solutions..................................................................................... 13
Phase Portraits .................................................................................................................................. 14
Determinants and their Properties ................................................................................................... 15
The Inverse of a Matrix...................................................................................................................... 16
Cramer’s Rule .................................................................................................................................... 17
Week 5................................................................................................................................................... 18
Linear Spaces and Linear Subspaces ................................................................................................. 18
Bases and Dimension of a Linear Space ............................................................................................ 18
Row Spaces ........................................................................................................................................ 19
Column Spaces .................................................................................................................................. 19
Solving Systems of Linear Equations ................................................................................................. 20
Week 6................................................................................................................................................... 21
Null Spaces ........................................................................................................................................ 21
Affine Subspaces ............................................................................................................................... 22
Linear Basis Transformations ............................................................................................................ 22
2
, Week 1
Eigenvectors and Eigenvalues
- Canonical basis vectors in ℝ𝑛 :
1 0 0
0 1 0
- 𝑒1 = ( ) , 𝑒2 = ( ),…, 𝑒1 = ( ).
⋮ ⋮ ⋮
0 0 1
- Identity matrix in 𝑅 𝑛×𝑛 :
1 0 ⋯ 0
0 1 ⋯ 0
- 𝐼𝑛 = ( ).
0 0 ⋱ ⋮
0 0 ⋮ 1
- An eigenvalue of a square matrix 𝐴 is a number 𝜆 ∈ ℂ such that the matrix 𝐴 − 𝜆𝐼𝑛 is
singular.
- The number 𝜆 ∈ ℂ is an eigenvalue of 𝐴 iff det(𝐴 − 𝜆𝐼) = 0. This is the characteristic
equation of matrix 𝐴.
- The trace of a square matrix 𝐴 is the sum of its diagonal entries.
▪ 𝜆1 + 𝜆2 + ⋯ + 𝜆𝑛 = trace(𝐴).
▪ 𝜆1 ∗ 𝜆2 ∗ ⋯ ∗ 𝜆𝑛 = det(𝐴).
- A vector 𝑣 ≠ 0 such that (𝐴 − 𝜆𝐼)𝑣 = 0 for some eigenvalue 𝜆 of 𝐴 is called an eigenvector
of 𝐴 corresponding to 𝜆.
- Note that: (𝐴 − 𝜆𝐼)𝑣 = 0 ⇔ 𝐴𝑣 = 𝜆𝑣.
Diagonalizable Matrices
- Let 𝐴 be a square matrix and let 𝜆1 , … , 𝜆2 , … , 𝜆𝑛 be its eigenvalues and 𝑣1 , 𝑣2 , … , 𝑣𝑛
corresponding eigenvectors, with 𝑃 ≔ [𝑣1 … 𝑣𝑛 ]. Then:
- If P is invertible, then:
𝜆1 0 ⋯ 0
0 𝜆2 ⋯ 0
▪ 𝑃−1 𝐴𝑃 = ( ).
0 0 ⋱ ⋮
0 0 ⋮ 𝜆𝑛
−1
- If 𝑄 𝐴𝑄 is a diagonal matrix 𝐷, then the columns of 𝑄 are eigenvectors of 𝐴 and the
diagonal entries of 𝐷 are eigenvalues of 𝐴.
Difference Equations
- One can use difference equations to model dynamical problems.
- If 𝐴 is a square matrix, then 𝑧𝑡+1 = 𝐴𝑧𝑡 is a system of linear difference equations (LDE).
- If you know the initial value 𝑧0 then you can iteratively calculate 𝑧1 , 𝑧2 , etc.
- In general: 𝑧𝑡 = 𝐴𝑡 𝑧0 , 𝑡 ≥ 0.
- A solution is of an LDE is a sequence of vectors {𝑧𝑡 }∞
𝑡=𝑡0 that ‘fits’ the LDE.
- The general solution of an LDE is the set containing all solutions of the LDE.
- The sequence {0, 0, 0, … } is a solution of an LDE for every 𝐴 and is called the null solution of
the LDE.
3
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