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Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist elementary invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that

• $a_i \mid a_{i+1}$ for all $1 \leq i \leq n-1$, and
• D=QAS.

The matrix $D$ is called Smith normal form of $A$ and is unique. Obviously, this applies to both $\mathbb Z$ and $\mathbb R[t]$.

Question 1: Is there any analogue for $R= \mathbb Z[t]$? Is there any classification of matrices over $\mathbb Z[t]$ up to equivalence?

A related question is the following:

Question 2: Is there any classification of $n \times n$-matrices over $\mathbb Z$ up to conjugation?

(The relation comes from looking at the matrix $t\cdot 1_n - A$ for $A \in M_n \mathbb Z$. Then, classification of $A$ up to conjugation is the same as classification of $t\cdot 1_n-A \in M_n \mathbb Z[t]$ up to equivalence.)

An obvious first invariant is the characteristic polynomial. Even if the matrix is assumed to be symmetric, it is unclear to me what kind of additional information could be added.

In that respect I know of a theorem of Latimer and MacDuffee which says that if the characteristic polynomial $f$ of $A \in M_n \mathbb Z$ is irreducible, then conjugacy classes of integer matrices with the same characteristic polynomial are in bijection with $\mathbb Z[\alpha]$-ideal classes in ${\mathbb Q}(\alpha)$, where $\alpha$ is a root of $f$. However, this seems to depend on the irreducibility of $f$ and I do not know of an extension to the general case. (This is nicely explained in notes by Keith Konrad.)

Question 3: Are there any other positive results going in the direction of the Theorem of Latimer-MacDuffee? What if the characteristic polynomial is a product of two irreducible polynomials?

Related but maybe easier:

Question 4: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for all primes $p$ and in $M_n \mathbb C$?

And finally

Question 5: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for a fixed prime $p$ and in $M_n \mathbb C$?

(Of course, Question 4 and 5 can be answered by looking at the Jordan decomposition for each of the fields separately. However, the question is, can we do something more conceptual?)

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# Analogue of Smith normal form for matrices over $\mathbb Z[t]$

Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist elementary matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that

• $a_i \mid a_{i+1}$ for all $1 \leq i \leq n-1$, and
• D=QAS.

The matrix $D$ is called Smith normal form of $A$ and is unique. Obviously, this applies to both $\mathbb Z$ and $\mathbb R[t]$.

Question 1: Is there any analogue for $R= \mathbb Z[t]$? Is there any classification of matrices over $\mathbb Z[t]$ up to equivalence?

A related question is the following:

Question 2: Is there any classification of $n \times n$-matrices over $\mathbb Z$ up to conjugation?

(The relation comes from looking at the matrix $t\cdot 1_n - A$ for $A \in M_n \mathbb Z$. Then, classification of $A$ up to conjugation is the same as classification of $t\cdot 1_n-A \in M_n \mathbb Z[t]$ up to equivalence.)

An obvious first invariant is the characteristic polynomial. Even if the matrix is assumed to be symmetric, it is unclear to me what kind of additional information could be added.

In that respect I know of a theorem of Latimer and MacDuffee which says that if the characteristic polynomial $f$ of $A \in M_n \mathbb Z$ is irreducible, then conjugacy classes of integer matrices with the same characteristic polynomial are in bijection with $\mathbb Z[\alpha]$-ideal classes in ${\mathbb Q}(\alpha)$, where $\alpha$ is a root of $f$. However, this seems to depend on the irreducibility of $f$ and I do not know of an extension to the general case. (This is nicely explained in notes by Keith Konrad.)

Question 3: Are there any other positive results going in the direction of the Theorem of Latimer-MacDuffee? What if the characteristic polynomial is a product of two irreducible polynomials?

Related but maybe easier:

Question 4: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for all primes $p$ and in $M_n \mathbb C$?

And finally

Question 5: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for a fixed prime $p$ and in $M_n \mathbb C$?

(Of course, Question 4 and 5 can be answered by looking at the Jordan decomposition for each of the fields separately. However, the question is, can we do something more conceptual?)