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Classification, Parameter Estimation and State Estimation, 2nd Edition by David M. J. Tax, Dick de Ridder, Ferdinand van der Heijden, Yaobin Zou, Ming Feng, Guangzhu Xu, Bangjun Lei

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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 ℝN and ℂN. 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 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:

(B.1)numbered Display Equation

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:

(B.2)numbered Display Equation

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)numbered Display Equation

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

The scalar–matrix multiplication αH

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