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The discriminant argument shows that for for $n \times n$ matrices over any field $k$, the Zariski closure of the set of non-diagonalizable matrices is proper in $\mathbb{A}^{n^2}$ -- an irreducible algebraic variety -- and therefore of smaller dimension. (Presumably the set is already Zariski closed, but this may be more trouble than it's worth to establish.) Being contained in a proper algebraic subset of affine or projective space is a very strong and useful way of saying that a set is "small" (except in the case that $k$ is finite!): in particular, its complement is Zariski dense.

One can use this observation to reduce many theorems in linear algebra to the diagonalizable case, the idea being that any polynomial identity that holds on a Zariski-dense set of all $n \times n$ matrices must hold (by definition of the Zariski topology!) for all matrices.

As a very simple example, one can immediately deduce that the characteristic polynomials $AB$ and $BA$ coincide, because if $A$ is invertible, the matrices are similar.

With a bit more care, one can derive the entire theory of determinants and characteristic polynomials from such specialization arguments. In fact by purely algebraic means it is possible to reduce to the case of $k = \mathbb{R}$ (and thereby define the determinant in terms of change of volume, etc.).

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The discriminant argument shows that for for $n \times n$ matrices over any field $k$, the Zariski closure of the set of non-diagonalizable matrices is proper in $\mathbb{A}^{n^2}$ -- an irreducible algebraic variety -- and therefore of smaller dimension. (Presumably the set is already Zariski closed, but this may be more trouble than it's worth to establish.) Being contained in a proper algebraic subset of affine or projective space is a very strong and useful way of saying that a set is "small" (except in the case that $k$ is finite!): in particular, its complement is Zariski dense.

One can use this observation to reduce many theorems in linear algebra to the diagonalizable case, the idea being that any polynomial identity that holds on a Zariski-dense set of all $n \times n$ matrices must hold (by definition of the Zariski topology!) for all matrices.

As a very simple example, one can immediately deduce that the characteristic polynomials $AB$ and $BA$ coincide, because if $A$ is invertible, they the matrices are similar.

With a bit more care, one can derive the entire theory of determinants and characteristic polynomials from such specialization arguments. In fact by purely algebraic means it is possible to reduce to the case of $k = \mathbb{R}$!mathbb{R}$(and thereby define the determinant in terms of change of volume, etc.). 1 The discriminant argument shows that for for$n \times n$matrices over any field$k$, the Zariski closure of the set of non-diagonalizable matrices is proper in$\mathbb{A}^{n^2}$-- an irreducible algebraic variety -- and therefore of smaller dimension. (Presumably the set is already Zariski closed, but this may be more trouble than it's worth to establish.) Being contained in a proper algebraic subset of affine or projective space is a very strong and useful way of saying that a set is "small" (except in the case that$k$is finite!): in particular, its complement is Zariski dense. One can use this observation to reduce many theorems in linear algebra to the diagonalizable case, the idea being that any polynomial identity that holds on a Zariski-dense set of all$n \times n$matrices must hold (by definition of the Zariski topology!) for all matrices. As a very simple example, one can immediately deduce that the characteristic polynomials$AB$and$BA$coincide, because if$A$is invertible, they are similar. With a bit more care, one can derive the entire theory of determinants and characteristic polynomials from such specialization arguments. In fact by purely algebraic means it is possible to reduce to the case of$k = \mathbb{R}\$!