Anon

396 The Basics of financial economeTrics

of the matrix A. The adjoint of the matrix A, denoted as Adj(A), is the fol-
lowing matrix:

()=

α⋅α⋅α

⋅⋅⋅⋅⋅

α⋅α⋅α

⋅⋅⋅⋅⋅

α⋅α⋅α





      





      

=

α⋅α⋅α

⋅⋅⋅⋅⋅

α⋅α⋅α

⋅⋅⋅⋅⋅

α⋅α⋅α





      





      

AdjA

jn

iijin

nnjnn

T

n

iini

nnnn

1,11,1,

,1 ,,

,1 ,,

1,12,1 ,1

1, 2, ,

1, 2, ,

The adjoint of a matrix A is therefore the transpose of the matrix obtained
by replacing the elements of A with their cofactors.
If the matrix A is nonsingular, and therefore admits an inverse, it can be
demonstrated that

()

A− =

A

A

1 Adj

A square matrix of order n, A, is said to be orthogonal if the following
property holds:

AA''==AA In

Because in this case A must be of full rank, the transpose of an orthogonal
matrix coincides with its inverse: AA−^1 = '.

Eigenvalues and Eigenvectors

Consider a square matrix A of order n and the set of all n-dimensional vec-
tors. The matrix A is a linear operator on the space of vectors. This means
that A operates on each vector producing another vector subject to the fol-
lowing restriction:

Ax()ab+=yAabxA+ y

Consider now the set of vectors x such that the following property holds:

Ax=λx

Any vector such that the above property holds is called an eigenvector of the
matrix A and the corresponding value of λ is called an eigenvalue.