ABSTRACT
This chapter is about the history of the algebra underpinning those methods of data analysis that depend on a strong appeal to visualization. It is impossible to do justice to this end in the limited space we have here, so we shall just mention the milestones. For fuller information see the excellent book of Macduffee (1946) and the paper of Stewart (1993). In fact, the algebra came long before its use in data analysis, its antecedents stretching back into history. However, it was only toward the end of the 18th century that interest began to expand in problems involving sets of linear equations, quadratic forms, and bilinear forms. These were in response to problems arising in applied mathematics, especially those concerned with geodetics and the conical forms of planetary motion. Famously, Gauss derived the least-squares equations for estimating the parameters (coordinates) of linear systems and an algorithm for solving the equations. Legendre (see, e.g., Gantmacher,
CONTENTS
2.1 Canonical Forms .......................................................................................... 18 2.1.1 Diagonalization of a Symmetric Matrix ....................................... 19 2.1.2 Decomposition of a Square Matrix ................................................ 19 2.1.3 Simultaneous Diagonalization of Two Symmetric Matrices
(Two-Sided Eigendecomposition) .................................................. 20 2.1.4 Results for Orthogonal Matrices, Skew-Symmetric
Matrices, and Hermitian Matrices................................................. 20 2.1.5 Singular Value Decomposition of a Rectangular Matrix ...........22
