Invariant Factors and Eigenvalues of a Linear Transform

Assume $F$ contains all eigenvalues of $T$, and demonstrate that each invariant factor can be factored into powers of linear polynomials where the linear factors correspond to eigenvalues of $T$.

In linear algebra, particularly when dealing with linear transformations and matrices, the concept of invariant factors provides a powerful way to understand the structure of a matrix. Invariant factors are related to the Smith normal form of a matrix, which is a diagonal matrix that can be reached by permissible row and column operations. These systematic rearrangements provide insights into the modules over a principal ideal domain. For a transformation represented by a matrix, invariant factors indicate how the vector space decomposes with respect to the action of the transformation.

Given the polynomial nature of eigenvalues of a matrix or linear transformation, this problem deals with understanding how these eigenvalues control the structure of invariant factors. The transformation's eigenvalues, which satisfy the characteristic equation, play a crucial role since each invariant factor is a divisor of the previous one, ultimately linking back to these eigenvalues. The fact that invariant factors can be expressed in terms of powers of linear polynomials with linear factors aligned with the eigenvalues of the transformation highlights the interplay between algebraic and geometric perspectives of matrices.

When tackling this type of problem, one should focus on the minimal polynomial and the characteristic polynomial of the transformation to understand its eigenvalue structure. The minimal polynomial's role is essential, as it captures the smallest-degree monic polynomial for which the transformation satisfies. When the field is algebraically closed, this polynomial splits into linear factors corresponding directly to the eigenvalues. Therefore, analyzing how invariant factors divide one another and relate back to these polynomials can deepen understanding of a matrix's structure through its eigenvalues.