Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 3Miscellaneous matrix results

1 INTRODUCTION

In this final chapter of Part One we shall discuss some more specialized topics, which will be applied later in this book. These include some further results on adjoint matrices (Sections 3.2 and 3.3), Hadamard products (Section 3.6), the commutation and the duplication matrix (Sections 3.7–3.10), and some results on the bordered Gramian matrix with applications to the solution of certain matrix equations (Sections 3.13 and 3.14).

2 THE ADJOINT MATRIX

We recall from Section 1.9 that the cofactor c_ij of the element a_ij of any square matrix A is (−1)^i+j times the determinant of the submatrix obtained from A by deleting row i and column j. The matrix C = (c_ij) is called the cofactor matrix of A. The transpose of C is called the adjoint matrix of A and we use the notation

(1)

We also recall the following two properties:

(2)

(3)

Let us now prove some further properties of the adjoint matrix.

Theorem 3.1

Let A be an n × n matrix (n ≥ 2), and let A^# be the adjoint matrix of A. Then,

1. (a) if r(A) = n, then
   (4)
2. (b) if r(A) = n − 1, then ...