ABSTRACT
Each square matrix A has a determinant; though the determinant can be used in the elementary study of the rank of a matrix and in solution of simultaneous linear equations, its most essential application in matrix theory is to the definition of the characteristic polynomial of a matrix. This chapter defines determinants, examines their geometric properties, and shows the relation of the characteristic polynomial of a matrix A to its characteristic roots (eigenvalues). These concepts will then be applied to the study of canonical forms for matrices under similarity. The characteristic equation, and hence its roots, is uniquely determined by A. This proves the essential uniqueness of the diagonal form—and gives a direct way to compute the coefficients. The construction of canonical forms for a matrix under similarity depends upon the study of the polynomial equations satisfied by the matrix or by the corresponding transformation.
