Final_1.pdf

APPENDIX: EIGENVALUE DECOMPOSITION

Consider a scalar land a corresponding vector v. They are an eigenvalue,
eigenvector pair of a square matrix Aif they satisfy the equation

Av=lv

The equation means that the vector vis special with respect to the matrix A.
Multiplying the vector with the matrix does not change the direction or ori-
entation of the vector. Its magnitude, however, is multiplied by the scalar l.
A square n×nmatrix may have neigenvaluesl 1 ,l 2 ,...,lnandncorrespon-
ding eigenvectors v 1 ,v 2 ,...,vn
Therefore

Av 1 =l 1 v 1

Av 2 =lv 2

M

Avn=lvn

Let us write this in matrix form. We construct a diagonal matrix Dwith the
eigenvalues forming the diagonal and a matrix Uwith each column corre-
sponding to an eigenvectors. Then

AU=UD

or

A=UDU–1

Note that we now have the matrix Ain terms of its eigenvalues and eigen-
vectors. This is the eigenvalue decomposition of A. The covariance matrix F
has special properties, and in that case

U–1=UT

and

F=UDUT

Pairs Selection in Equity Markets 103