Chapter V. Theory of a Single Linear Transformation

Abstract

This goal of this chapter is to find finitely many canonical representatives of each similarity class of square matrices with entries in a field and correspondingly of each isomorphism class of linear maps from a finite-dimensional vector space to itself.

Section 1 frames the problem in more detail. Section 2 develops the theory of determinants over a commutative ring with identity in order to be able to work easily with characteristic polynomials $\det(X I-A)$. The discussion is built around the principle of “permanence of identities,” which allows for passage from certain identities with integer coefficients to identities with coefficients in the ring in question.

Section 3 introduces the minimal polynomial of a square matrix or linear map. The Cayley–Hamilton Theorem establishes that such a matrix satisfies its characteristic equation, and it follows that the minimal polynomial divides the characteristic polynomial. It is proved that a matrix is similar to a diagonal matrix if and only if its minimal polynomial is the product of distinct factors of degree 1. In combination with the fact that two diagonal matrices are similar if and only if their diagonal entries are permutations of one another, this result solves the canonical-form problem for matrices whose minimal polynomial is the product of distinct factors of degree 1.

Section 4 introduces general projection operators from a vector space to itself and relates them to vector-space direct-sum decompositions with finitely many summands. The summands of a direct-sum decomposition are invariant under a linear map if and only if the linear map commutes with each of the projections associated to the direct-sum decomposition.

Section 5 concerns the Primary Decomposition Theorem, whose subject is the operation of a linear map $L:V\to V$ with $V$ finite-dimensional. The statement is that if $L$ has minimal polynomial $P_1(X)^{l_1}\cdots P_k(X)^{l_k}$ with the $P_j(X)$ distinct monic prime, then $V$ has a unique direct-sum decomposition in which the respective summands are the kernels of the linear maps $P_j(L)^{l_j}$, and moreover the minimal polynomial of the restriction of $L$ to the $j^\mathrm{th}$ summand is $P_j(X)^{l_j}$.

Sections 6–7 concern Jordan canonical form. For the case that the prime factors of the minimal polynomial of a square matrix all have degree 1, the main theorem gives a canonical form under similarity, saying that a given matrix is similar to one in “Jordan form” and that the Jordan form is completely determined up to permutation of the constituent blocks. The theorem applies to all square matrices if the field is algebraically closed, as is the case for $\mathbb C$. The theorem is stated and proved in Section 6, and Section 7 shows how to make computations in two different ways.