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Summary Linear Algebra - Econometrics
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  • Course
  • Linear Algebra
  • Institution
  • Rijksuniversiteit Groningen (RuG)
  • Book
  • Mathematics for Economists

Dit is een samenvatting voor het vak Linear Algebra voor de studie Econometrics and Operations Research. Het bevat voorbeelden en zowel de hoofdstukken uit het boek als de lectures zijn erin samengevat.

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  • No
  • H23, 24, 25, 26, 27
  • June 9, 2017
  • 20
  • 2016/2017
  • Summary

Subjects

  • econometrics
  • econometrie
  • linear
  • algebra
  • eor

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  • Rijksuniversiteit Groningen (RuG)
  • Econometrics and Operations Research
  • Linear Algebra
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Chapter 23
- Paragraph 1 – Definitions and examples
Let A be a square matrix.
An eigenvalue of A = a number r, which when subtracted from each of the diagonal entries of A,
converts A into a singular matrix
Note: a matrix is singular if and only if the determinant of the matrix is zero

r is an eigenvalue of 𝐴 if and only if 𝐴 – 𝑟𝐼 is a singular matrix:
det(𝐴 – 𝑟𝐼) = 0.
For a n x n matrix A, the left-hand side of the equation is an nth order polynomial in the variable r,
called the characteristic polynomial of A. An nth order polynomial has at most n roots and exactly n
roots if one counts roots with their multiplicity and complex roots.
So, an n x n matrix has at most n eigenvalues.

Theorem 23.1
The diagonal entries of a diagonal matrix D are the eigenvalues of D

Theorem 23.2
A square matrix A is singular if and only if 0 is an eigenvalue of A

A matrix M whose entries are nonnegative and whose columns (or rows) each add to 1 is called a
1 2

Markov matrix, e.j. (43 3
1). If we subtract a 1 from each diagonal entry of the Markov matrix, then
4 3
3 2
−4 3
each column of the transformed matrix adds up to 0, M − 1𝐼 = ( 3 2). If the columns of a
−3
4
square matrix add up to (0, …, 0) the rows are linearly dependent and the matrix must be singular.
So, r = 1 is an eigenvalue of every Markov matrix.

Recall: a square matrix B is non-singular if and only if the only solution of Bx = 0 is x = 0.
And B is singular if and only if the system Bx = 0 has a nonzero solution.
The fact that the square matrix 𝐴 – 𝑟𝐼 is a singular matrix when r is an eigenvalue of A means that
the system of equations (𝐴 – 𝑟𝐼)𝒗 = 𝟎 has a solution different from v = 0.
When r is an eigenvalue of A, a nonzero vector v such that (𝐴 – 𝑟𝐼)𝒗 = 𝟎 is called an eigenvector of
A corresponding to the eigenvalue r.
Theorem 23.3
Let A be an n x n matrix and let r be a scalar. Then, the following statements are equivalent:
1) Subtracting r from each diagonal entry of A transforms A into a singular matrix
2) 𝐴 – 𝑟𝐼 is a singular matrix
3) det(𝐴 – 𝑟𝐼) = 0
4) (𝐴 – 𝑟𝐼)𝒗 = 𝟎 for some nonzero vector v
5) 𝐴𝒗 = 𝑟𝒗 for some nonzero vector v

The set of all solutions to (𝐴 – 𝑟𝐼)𝒗 = 𝟎, including v = 0, is called the eigenspace of A with respect to
the corresponding eigenvalue.
Note: the eigenvalues of an upper- or lower-triangular matrix are precisely its diagonal entries
Note: for an invertible matrix A, r is an eigenvalue of A if and only if 1/r is an eigenvalue of 𝐴−1
Note: 𝐴−1 𝐴 = 𝐼

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