Daniel J. Duffy - Numerical Methods in Computational Finance

We give a precise overview of the operations that can be performed on matrices. We mention that they subsume operations on one-dimensional vectors because the latter can be viewed as matrices with one column (or with one row, depending on the context).

The notation and operations for vectors are:

(5.13)

and for rectangular matrices:

(5.14)

Some special cases of matrices are:

Row matrix: has one row (row vector).

Column matrix: has one column (column vector).

Zero matrix: all entries have the value zero.

Diagonal matrix: all entries zero except those on main diagonal.

Identity matrix: diagonal matrix all of whose diagonal elements == 1.

Matrix operations are:

(5.15)

Matrix addition:

Scalar multiplication:

Matrix multiplication:

Transpose of a matrix AA new matrix obtained by writing the rows of A as columns.Applicable to rectangular matrices:

1 Matrix trace (the sum of the diagonal element of a square matrix):

An example of matrix multiplication is:

In this section we classify matrices based on some fundamental (computed) property. These properties are used in various areas of numerical analysis and its applications. It is then a useful reference to group them as we do here. In general, matrices can have real or complex values. In the latter case we recall complex conjugation :

In many cases we are concerned with square matrices with real values.

A nilpotent matrix A is a square matrix such that for some positive integer p . It is said to be of index p if p the least positive integer for which . For example, the matrix:

is nilpotent with index 2. More generally, a triangular matrix of size n with zeros along the main diagonal is nilpotent with index . For example, the follow matrix is nilpotent with index 3:

The determinant and trace of a nilpotent matrix are always zero. Thus, such matrices are not invertible. However, and are invertible where N is a nilpotent matrix:

(5.16)

where I is the identity matrix. Since there are only finitely many non-zero terms, we see that both sums converge.

An idempotent matrix A is one for which . Examples are:

Idempotent matrices arise in regression analysis and econometrics, for example in ordinary least squares problems, in particular when estimating sums of squared residuals.

An involutory matrix is one that is its own inverse:

An example is:

For example, the Pauli matrices are involutory:

We introduce the important class of normal matrices by introducing some prerequisite notation. The transpose of a real matrix is an matrix formed by exchanging the rows and columns of A :

In the case of a complex matrix A , the Hermitian transpose is the complex conjugate transpose of A :

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:

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.