Classification, Parameter Estimation and State Estimation, 2nd Edition

BTopics Selected from Linear Algebra and Matrix Theory

Whereas Appendix A deals with general linear spaces and linear operators, the current appendix restricts attention to linear spaces with finite dimension, that is ℝ^N and ℂ^N. Therefore, all that has been said in Appendix A also holds true for the topics of this appendix.

B.1 Vectors and Matrices

Vectors in ℝ^N and ℂ^N 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:

(B.1)

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 h_n,m (the elements) on an orthogonal grid of N rows and M columns:

(B.2)

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:

(B.3)

Of course, a matrix can also be regarded as a vertical arrangement of N row vectors.

The scalar–matrix multiplication αH