Numerical Methods in Computational Finance - Daniel J. Duffy

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j less-than-or-equal-to n 2nd Row upper A Superscript upper H Baseline equals left-parenthesis upper A overbar right-parenthesis Superscript down-tack Baseline equals upper A overbar Superscript down-tack Baseline Hermitian transpose period EndLayout"/>

      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:

upper A equals Start 2 By 2 Matrix 1st Row 1st Column 2 2nd Column 3 plus 4 i 2nd Row 1st Column 3 minus 4 i 2nd Column negative 5 EndMatrix comma upper A overbar equals Start 2 By 2 Matrix 1st Row 1st Column 2 2nd Column 3 minus 4 i 2nd Row 1st Column 3 plus 4 i 2nd Column negative 5 EndMatrix period

      5.6.3 Unitary and Orthogonal Matrices

upper U Superscript negative 1 Baseline equals upper U Superscript upper H Baseline equals upper U overbar Superscript down-tack Baseline period

      Unitary matrices are normal: italic upper U upper U Superscript upper H Baseline equals italic upper U upper U Superscript negative 1 Baseline equals upper I equals upper U Superscript negative 1 Baseline upper U equals upper U Superscript upper H Baseline upper U period Furthermore, given two vectors, multiplication by U preserves inner products left-parenthesis upper U x comma upper U y right-parenthesis equals left-parenthesis x comma y right-parenthesis. 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)upper Q Superscript down-tack Baseline upper Q equals italic upper Q upper Q Superscript down-tack Baseline equals upper I or upper Q Superscript down-tack Baseline equals upper Q Superscript negative 1 Baseline period

      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 n times n matrix upper A is positive definite if:

      (5.18)left-parenthesis italic upper A x comma x right-parenthesis greater-than 0 for-all x element-of normal double struck upper R Superscript n

      and positive semidefinite if:

      (5.19)left-parenthesis italic upper A x comma x right-parenthesis greater-than-or-equal-to 0 for-all x element-of normal double struck upper R Superscript n Baseline period

      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.

       

 upper A equals Start 3 By 3 Matrix 1st Row 1st Column 2 2nd Column negative 1 3rd Column 0 2nd Row 1st Column negative 1 2nd Column 2 3rd Column negative 1 3rd Row 1st Column 0 2nd Column negative 1 3rd Column 2 EndMatrix period

      To prove this let z equals left-parenthesis a comma b comma c right-parenthesis Superscript down-tack. Then

left-parenthesis italic upper A z comma z right-parenthesis identical-to z Superscript down-tack Baseline italic upper A z equals a squared plus left-parenthesis a minus b right-parenthesis squared plus left-parenthesis b minus c right-parenthesis squared plus c squared greater-than 0 period

      We can take the square root upper A Superscript 1 slash 2 of a positive definite matrix A, and it is a well-defined function. Furthermore, we can factor a positive definite matrix A into:

      5.6.5 Non-Negative Matrices

      A matrix A is non-negative (written upper A greater-than-or-equal-to 0) if all its elements are real and non-negative. A matrix A is greater than a matrix B if upper A minus upper B 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:

italic upper P upper A upper P Superscript down-tack Baseline equals Start 2 By 2 Matrix 1st Row 1st Column upper A 11 2nd Column upper A 12 2nd Row 1st Column 0 2nd Column upper A 22 EndMatrix comma left-parenthesis upper A 11 comma upper A 12 comma upper A 22 a r e square matrices right-parenthesis period

      The matrix A is called irreducible if no

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