Numerical Methods in Computational Finance. Daniel J. Duffy

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      We are ready to give some more definitions of special kinds of matrices:

       Normal:

       Hermitian:

       Skew-Hermitian:

       Symmetric (real) matrix:

      An example of a Hermitian (and hence normal matrix) is:

      5.6.3 Unitary and Orthogonal Matrices

A matrix U is unitary if its inverse equals its Hermitian transpose:

      Unitary matrices are normal: Furthermore, given two vectors, multiplication by U preserves inner products . Unitary matrices are important in quantum mechanics because they preserve norms and thus probability amplitudes. See the Appendix in this chapter.

      A real square matrix Q is orthogonal if:

      (5.17)

      Orthogonal matrices are important in numerical linear algebra applications because of their numeric stability properties. Some application areas are matrix decomposition methods (Golub and van Loan (1996)) such as:

       QR decomposition orthogonal, upper triangular.

       Singular Value Decomposition (SVD) and V orthogonal, diagonal matrix.

       Eigendecomposition symmetric, Q orthogonal, diagonal.

       Polar decomposition where is orthogonal, symmetric positive-semidefinite.

      A particular application area is solving overdetermined systems of linear equations and solving ill-posed linear systems.

      5.6.4 Positive Definite Matrices

      This is another very important class of matrices. Matrices having this property are highly desirable in applications and algorithms. An matrix is positive definite if:

      (5.18)

      and positive semidefinite if:

      (5.19)

      Some necessary conditions for positive semidefiniteness are:

       The diagonal elements of A must be positive.

       A is positive definite if and only if all its eigenvalues are positive.

       The element of A having the greatest absolute value must be on the diagonal of A.

       

An example of a positive definite matrix A is:

      To prove this let . Then

      We can take the square root of a positive definite matrix A, and it is a well-defined function. Furthermore, we can factor a positive definite matrix A into:

(5.20)

where L is a lower triangular matrix having positive values on its diagonal. Equation (5.20) is called the Cholesky decomposition for A.

      5.6.5 Non-Negative Matrices

      A matrix A is non-negative (written ) if all its elements are real and non-negative. A matrix A is greater than a matrix B if is positive. These kinds of matrices have many applications, for example Markov chains and stochastic matrix theory. They are also relevant when we construct monotone finite difference schemes and M-matrices to approximate the solution of differential equations, as we shall see later in this book.

      5.6.6 Irreducible Matrices

      A matrix A is said to be reducible if there exists a permutation matrix P such that:

      The matrix A is called irreducible if no