Linear Algebra and Matrix Methods in Econometrics
Henri Theil
University of Florida

Contents
1 Introduction 1

2 Why are matrix methods useful in econometrics? 2
2.1 Linear systems and quadratic forms . . . . . . . . . . . . . . . 2
2.2 Vectors and matrices in statistical theory . . . . . . . . . . . . 4
2.3 Least squares in the standard linear model . . . . . . . . . . . 5
2.4 Vectors and matrices in consumption theory . . . . . . . . . . 7

3 Partitioned matrices 10
3.1 The algebra of partitioned matrices . . . . . . . . . . . . . . . 10
3.2 Block-recursive systems . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Income and price derivatives revisited . . . . . . . . . . . . . . 12

4 Kronecker products and the vectorization of matrices 14
4.1 The algebra of Kronecker products . . . . . . . . . . . . . . . 14
4.2 Joint generalized least-squares estimation of several equations 15
4.3 Vectorization of matrices . . . . . . . . . . . . . . . . . . . . . 17

5 Differential demand and supply systems 19
5.1 A differential consumer demand system . . . . . . . . . . . . . 19
5.2 A comparison with simultaneous equation systems . . . . . . . 21
5.3 An extension to the inputs of a firm: A singularity problem . . 22
5.4 A differential input demand system . . . . . . . . . . . . . . . 23
5.5 Allocation systems . . . . . . . . . . . . . . . . . . . . . . . . 24
5.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Definite and semidefinite square matrices 27
6.1 Covariance matrices and Gauss-Markov further considered . . 27
6.2 Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Block-diagonal definite matrices . . . . . . . . . . . . . . . . . 30

7 Diagonalizations 31
7.1 The standard diagonalization of a square matrix . . . . . . . . 31
7.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.3 Aitken’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.4 The Cholesky decomposition . . . . . . . . . . . . . . . . . . . 35
7.5 Vectors written as diagonal matrices . . . . . . . . . . . . . . 35
7.6 A simultaneous diagonalization of two square matrices . . . . 36
7.7 Latent roots of an asymmetric matrix . . . . . . . . . . . . . . 37

8 Principal components and extensions 40
8.1 Principal components . . . . . . . . . . . . . . . . . . . . . . . 40
8.2 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.3 Further discussion of principal components . . . . . . . . . . . 42
8.4 The independence transformation in microeconomic theory . . 43
8.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.6 A principal component interpretation . . . . . . . . . . . . . . 47

9 The modeling of a disturbance covariance matrix 49
9.1 Rational random behavior . . . . . . . . . . . . . . . . . . . . 49
9.2 The asymptotics of rational random behavior . . . . . . . . . 50
9.3 Applications to demand and supply . . . . . . . . . . . . . . . 53

10 The Moore-Penrose inverse 56
10.1 Proof of the existence and uniqueness . . . . . . . . . . . . . . 56
10.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.3 A generalization of Aitken’s theorem . . . . . . . . . . . . . . 58
10.4 Deleting an equation from an allocation model . . . . . . . . . 62

A Linear independence and related topics 64
B The independence transformation 66
C Rational random behavior 70