Managing Credit Risk in Corporate Bond Portfolios : A Practitioner's ...

Mathematical Preliminaries 15denoted I. The inverse of the matrix A is denoted A 1 . A necessary conditionfor a matrix to be invertible is that all its column vectors are linearlyindependent.In the special case where the transpose of a matrix is equal to theinverse of a matrix, that is, A T A 1 , the matrix is referred to as anorthogonal matrix.Eigenvalues and EigenvectorsThe eigenvalues of a square matrix A are real or complex numbers suchthat the vector equation Ax x has nontrivial solutions. The correspondingvectors x0 are referred to as the eigenvectors of A. Any n nmatrix has n eigenvalues, and associated with each eigenvalue is a correspondingeigenvector. It is possible that for some matrices not all eigenvaluesand eigenvectors are distinct. The sum of the n eigenvalues equals thesum of the entries on the diagonal of the matrix A, called the trace of A.Thus,trace A ani1a ii anl ii1If 0 is an eigenvalue of the matrix, the matrix is referred to as a singularmatrix. Matrices that are singular do not have an inverse.Diagonalization of a MatrixWhen x is an eigenvector of the matrix A, the product Ax is equivalent tothe multiplication of the vector x by a scalar quantity. This scalar quantityhappens to be the eigenvalue of the matrix. One can conjecture from thisthat a matrix can be turned into a diagonal matrix by using eigenvectorsappropriately. In particular, if the columns of matrix M are formed usingthe eigenvectors of A, then the matrix operation M 1 AM is a diagonalmatrix with eigenvalues of A as the diagonal elements. However, for this tobe true, the matrix M must be invertible. Stated differently, the eigenvectorsof the matrix A must form a set of linearly independent vectors.It is useful to remark here that any matrix operation of the type B 1 ABwhere B is an invertible matrix is referred to as a similarity transformation.Under a similarity transformation, eigenvalues remain unchanged.Properties of Symmetric MatricesSymmetric matrices have the property that all eigenvalues are real numbers.If, in addition, the eigenvalues are all positive, then the matrix is referred to