In Linear Algebra, we consider characteristic polynomials.
Is a characteristic polynomial we consider in Linear Algebra a polynomial or a polynomial function?
I think it is a polynomial function.
I am reading "Introduction to Linear Algebra" (in Japanese) by Kazuo Matsuzaka.
In this book, the characteristic polynomial of a linear map $F$ is defined by $\det(A - \lambda I)$, where $A$ is a matrix which represents $F$.
And in this book, the author defines a determinant only for a matrix whose elements belong to a some field $K$.
If $\det(A - \lambda I)$ is a polynomial, then the elements of $A - \lambda I$ are polynomials too. But the author didn't define a determinant for a matrix whose elements are polynomials.