## An Introduction to the |

This book has been written with the object of giving an account of the various ways in which matrices of finite order can be reduced to canonical form under different important types of transformation. While the work has been planned to serve as a sequel to a former publication, *The Theory of Determinants, Matrices, and Invariants* (1928), circumstances have allowed us to make it practically independent and self-contained, with the least possible overlapping of material in the two books. A certain knowledge of the elementary theory of determinants is presupposed, but no previous acquaintance with matrices.

The volume on *Invariants* - as it will be referred to in subsequent pages - in giving an introductory account of matrices and determinants, treated only of such properties as belonged to the general linear transformation; for these are the properties which have the most direct bearing on the projective invariant theory, to which the later chapters were devoted. In the nomenclature of the work before us, the treatment was confined to the diagonal case of the classical canonical form, in which the elementary divisors are necessarily linear.

In the present work we return to consider, in close detail, those important cases in which the elementary divisors are no longer restricted to be linear, but may be of general degree. To adopt a geometrical mode of speaking, it is as if we had formerly been concerned purely with the projective, properties of quadrics in general position, but had now returned to the consideration of all possible distinctions between quadrics under certain prescribed conditions; such distinctions, for example, as those which persist through all projective transformations, or again through all rotations, and so on.

The subject-matter of the canonical reduction of matrices, which has numerous and important applications, has received attention in several treatises and a large number of original papers. The historical notes which we have appended to each chapter are intended to give a brief review of what has been done on each topic, to apportion due credit to pioneers, and to stimulate the, student to further reading. (We would warn him, however, to make sure at the outset, in reading any work on groups or matrices, whether the author means AB or what we have denoted by BA when he writes a product.) The most complete accounts of the theory available are those of Muth *(Elementarteiler,* 1899) and Cullis *(Matrices and Determinoids,* Vols. I, II, Ill, 1913, 1918, 1925). We have preferred to follow the lead of Cullis, who develops the theory in terms of the structure and properties of matrices - in matrix idiom, as it were, rather than in terms of bilinear and quadratic forms, or of linear substitutions.

We take the opportunity of acknowledging our indebtedness to the work of those writers who have given a sustained account of the theory, in one guise or another; in particular to Muth, as above, to Bromwich (Cambridge Tract on *Quadratic Forms,* 1906, and various papers), to Bôcher *(Higher Algebra,* 1907), Hilton *(Linear Substitutions,* 1914), Cullis (Vol. Ill of *Matrices and Determinoids,* 1925), and Dickson *(Modern Algebraic Theories,* 1926).

While we have tried to include all the principal features of the theory and have sought to make the sequence of argument reasonably fluent, even allowing ourselves moderate latitude in digression and explanation, we have, at the same time, aimed at a certain compactness in the formulae and demonstrations. This has been achieved in the first place by a systematic use of the matrix notation, to which we shall again refer; in the second place, by confining the contents of each chapter almost entirely to general theorems, and by relegating corollaries and applications to the interspersed sets of examples. These examples are intended to serve not so much as exercises, many being quite easy, but rather as points of relaxation, and running commentary; they will, however, be found to contain many well-known and important theorems, which the notation establishes in the minimum of space.

We attach the greatest importance to the choice of notation. Inferring from perusal of Cullis that the emphasis laid since the time of Cayley on the square matrix might well be removed, we resolved to continue the plan adopted in *Invariants* by making the fullest use of rectangular matrices and submatrices, and of partitioned matrices, by insisting on the condition that the non-commutative rules of product order hold without exception, and by distinguishing always between a matrix of a single row and one of a single column. When this is done, all the systems which appear, whether scalars, vectors, or matrices, can be regarded as rectangular matrices or products of rectangular matrices, and the theory is thus greatly unified. We would draw special attention to the notation x'Ay for the bilinear form, x'Ax for the quadratic, and x(bar)'Ax for the Hermitian form, believing that these notations will enable the linear transformations and the bilinear, quadratic, and Hermitian forms which are fundamental, for example, in analytical geometry, dynamics, or mathematical statistics, to be manipulated with ease.

Through considerations of space we have not been able to include many applications to geometry, but the results are readily adaptable: nor to the theory of Groups, where, as Schur has shown, partitioned matrices can be used with elegance and advantage.

The reader already familiar with the theory will also observe that certain established methods of dealing with the subject have hardly been touched upon, notably the methods of Weierstrass and Darboux, the theory of regular minors of determinants and the treatment of quadratic forms by the methods of Kronecker. We have, in fact, allowed ourselves a free hand in dealing with the results of earlier writers, in the belief that the outcome would prove to be an easier approach to a subject that has often failed to win affection; and the methods of H J S Smith, Sylvester, Frobenius, and Dickson proved in themselves quite adequate without the inclusion of other parallel theories. A thorough assimilation of the algebraic implications of Euclid's H.C.F. process, and of the notion of linear dependence, furnishes the clue to many passages. Our tribute to Kronecker finds expression in Chapter IX, which is an essay towards giving a fresh derivation of his classical results concerning singular pencils; we have treated this by rational methods, and we trust that an intricate argument has been materially simplified.

Our best thanks are due to Dr E T Copson and Mr D E Rutherford at St Andrews University, who have taken an interest in the progress of the work, and have offered valuable suggestions at the proof-reading stage; and especially to Dr John Dougall, for his critical vigilance and expert mathematical and technical help during the passage of the work through the press.

H W Turnbull

A C Aitken

St Andrews

Edinburgh

December, 1931.

JOC/EFR August 2007

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