# Similarity and normal forms

This post will prove the uniqueness of Jordan form, Smith normal form and Rational canonical form in an equivalence class of similar matrices. A criterion to check similarlity of matrices using these forms is given, and a nice lemma of similarity being irrelevant to field extension is shown using rational canonical form.

This post will prove the uniqueness of Jordan form, Smith normal form and Rational canonical form in an equivalence class of similar matrices. This is why they are called canonical.

Theorem 1Let be an matrix (over an algebraically closed field ). Then there exists a unique matrix among all the similar matrices of , that is in Jordan form. (up to permutation of Jordan blocks)

**Proof**

We have seen in a previous post that how determines the Jordan blocks. This determines the uniqueness.

Alternatively, notice that if we regard to be the matrix of a linear transformation , then a Jordan form corresopnds to the primary decomposition of finitely generated modules over PID, which is unique up to permutation.

Theorem 2Let be an matrix (over a PID). Then there exists a unique matrix among all the similar matrices of , that is in Smith normal form. (up to associatedness of elementary divisors)

**Proof**

Define to be the greatest common divisor of all minors of of order for .

Once we have put the matrix in its Smith normal form = , it is clear that

(if we set if ) Therefore the theorem is proved once we have

Lemma 1 (Invariance of)Let , where are invertible matrices. Then for any , .

**Proof** Clear from Binet-Cauchy formula.

Theorem 3Let be an matrix (over a field ). Then there exists a unique matrix among all the similar matrices of , that is in rational canonical form.

**Proof**

If we regard to be the matrix of a linear transformation , then rational canonical form corresopnds to the decomposition of finitely generated modules over PID, which is unique.

This gives us some methods to check when two matrices are similar.

Corollary 1Two matrices and over a field iff they have the same Jordan form/rational canonical form.

Smith normal form also helps determine the similarity of matrices.

Corollary 2Two matrices and over a field are similar iff and has the same Smith normal form.

**Proof**

Regard to be the matrices of linear transformations . Remember that the Smith normal form of () represents exactly the decomposition of -module. (-module) Therefore it suffices to show that , are similar iff the -module structure of is the same as its -module structure, which is obvious.

**Similarity and underlying field**

As Jordan form needs the field to be extended to its algebraic closure, it raises a natural question: Let are two fields, and are two matrices with entries in . If are similar over , are they similar over ? The answer is yes.

Theorem 4Let are two fields, and are two matrices with entries in . If are similar over , then they are similar over .

**Proof for infinite field case
**

is equivalent to = 0. Thus we are asking if is solvable over , can it be solved over .

Using Kronecker product, rewrite the equation as

This shows that the solution space of (over either or ) can be spanned by matrices with entries in . Let be one such basis, and consider the multinomial

is not identically 0, because over , there exists some such that . If the field is infinite, then this implies that for some , , meaning that is solvable over .

There is a swft argument for the general case, as indicated by loup blanc in this post.

**Proof 2**

1. , are similar if and only if they have the same rational canonical form, as proved above.

2. Notice furthermore that the rational canonical form of in is the same as that in . Reason:

Suppose that we can find invertible such that is in Smith normal form. By uniqueness of normal form, and that , the Smith normal form in is the same. This implies that the rational canonical form are the same.

If are similar over , they have the same rational canonical form over . But their rational canonical form over are the same, so they are similar over .

**Quick summary**

1. Jordan form/rational canonical form of a linear map is merely the matrix form of the structure of as a -module.

2. The similarity problem is the same as whether the -module structure of are the same.

3. is exactly the relations matrix/kernel of the natural map . Thus it is significant in rational canonical form/Jordan form.