Whereas Appendix A deals with general linear spaces and linear operators, the current appendix restricts attention to linear spaces with finite dimension, that is ℝN and ℂN. Therefore, all that has been said in Appendix A also holds true for the topics of this appendix.
Vectors in ℝN and ℂN are denoted by bold-faced letters, for example f and g. The elements in a vector are arranged either vertically (a column vector) or horizontally (a row vector). For example:
The superscript T is used to convert column vectors to row vectors. Vector addition and scalar multiplication are defined as in Section A.1.
A matrix H with dimension N × M is an arrangement of NM numbers hn,m (the elements) on an orthogonal grid of N rows and M columns:
The elements are real or complex. Vectors can be regarded as N × 1 matrices (column vectors) or 1 × M matrices (row vectors). A matrix can be regarded as a horizontal arrangement of M column vectors with dimension N, for example:
Of course, a matrix can also be regarded as a vertical arrangement of N row vectors.
The scalar–matrix multiplication αH