ABSTRACT
This chapter describes the classification of square matrices. It also describes a normal form for matrices under similarity and deals with the special case of matrices similar to a matrix in diagonal form. The chapter considers the general problem of transforming a matrix to diagonal form by similarity transformation. The place of orthogonal transformations is taken by unitary matrices. Quadratic forms are used in geometry to describe conics and quadrics, but they also occur very widely in mechanics, statistics, economics and elsewhere, so a reduction theory will have many uses. The chapter discusses a transformation of coordinates for which the matrix of the form becomes fairly simple, for example, diagonal. It shows that every symmetric matrix is congruent to a diagonal matrix.
