31 Aug A Problem in Linear Algebra
In this article we shall show that the characteristic polynomial of both and are the same, where and are matrices over , a ring with unity.
Let denote the characteristic polynomial of . Thus, we intend to show,
We first take and specialize in this ring. By multiplicity of the determinant function, we have that the characteristic polynomial of two similar matrices are the same.
We keep fixed and let be a diagonalizable (semisimple) matrix. Let be an invertible matrix such that is diagonal. Then,
Fact : The set of semisimple matrices is dense in .
By making use of the this and continuity of det function we conclude that , .
Let and be two matrices where and are variables. Then,
We call this polynomial as . Corresponding to this polynomial we have a function where,
As , we have that is the zero function. Thus, we have that is the zero polynomial.
As the identity, holds for entries in .
We make use of the following facts : –
- There is a unique homomorphism from where is a ring with unity.
- If is a homomorphism then there is a unique homomorphism which preserves action of on constants and .
Using the above facts there is a unique homomorphism
But as in
where and .