The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 147 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 147

are all zero except the elements on the diagonal that coincide with those

of the matrix A:

a 11 a 12 ··· a 1 n a 11 0 ··· 0

a 21 a 22 ··· a 2 n 0 a 22 ··· 0

A = · · · · ⇒ dgA = · · · ·

· · · · · · · ·

· · · · · · · ·

an 1 an 2 ··· ann 0 0 ··· ann

The trace of a square matrix A is the sum of its diagonal elements:

n

trA = ∑ aii

i = 1

A square matrix is called symmetric if the elements above the diago-

nal are equal to the corresponding elements below the diagonal: aij = aji.

A matrix is called skew-symmetric if the diagonal elements are zero and

the elements above the diagonal are the opposite of the corresponding

elements below the diagonal: aij = –aji, i ≠ j, aii = 0.

The most commonly used symmetric matrix in finance and econo-

metrics is the covariance matrix, also referred to as the variance-covari-

ance matrix. (See Chapter 6 for a detailed explanation of variances and

covariances.) For example, suppose that there are N risky assets and

that the variance of the excess return for each risky asset and the covari-

ances between each pair of risky assets are estimated. As the number of

credit risky assets is N there are N^2 elements, consisting of N variances

(along the diagonal) and N^2 – N covariances. Symmetry restrictions

reduce the number of independent elements. In fact the covariance σij(t)

between risky asset i and risky asset j will be equal to the covariance

between risky asset j and risky asset i. We can therefore arrange the

variances and covariances in the following square matrix V:

σ 11 , · σ 1 , i · σ 1 , N

· ··· ·

V = σ 1 , i · σii, · σiN,

· ··· ·

σ 1 , N · σiN, · σNN,

Notice that V is a symmetric matrix.