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Let $n$ be an integer $\ge2$.

(a) What is the least integer $k$ such that any $n$ by $n$ integer matrix is a $\mathbb Z$-linear combination of $k$ indempotents?

(The idempotents are also required to be $n$ by $n$ integer matrices.)

Denote the above integer $k$ by $k_n(\mathbb Z)$.

(b) Is the sequence $(k_n(\mathbb Z))$ bounded?

One can define $k_n(R)$ for any commutative ring $R$, and ask the same questions as above. One can then define $k_n$ as the largest of all the $k_n(R)$ (when $R$ runs over the class of all commutative rings).

If $R$ is a field, then the $k_n(R)$ have been computed in

Clément de Seguins Pazzis, On decomposing any matrix as a linear combination of three idempotents, Linear Algebra Appl. 433-4 (2010), 843-855, http://arxiv.org/abs/0907.4949.

Let $A=(a_{ij})_{1\le i,j\le n}$ be a family of indeterminates. In particular $A$ can be viewed as an $n$ by $n$ matrix with entries in the polynomial ring $$ S:=\mathbb Z[(a_{ij})_{1\le i,j\le n}]. $$ Then $k_n$ is also the least integer $k$ such that $A$ is an $S$-linear combination of $k$ indempotents.

We can ask about $k_n$ the same questions as we asked above about $k_n(\mathbb Z)$.

I haven't even been able to compute $k_2(\mathbb Z)$! The equality $$ \begin{pmatrix}a&b\\c&d\end{pmatrix}=(a-2)\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}1&b\\0&0\end{pmatrix}+\begin{pmatrix}1&0\\c&0\end{pmatrix}+d\begin{pmatrix}0&0\\0&1\end{pmatrix} $$ implies $k_2(\mathbb Z)\le k_2\le4$, so that we get $$ 2\le k_2(\mathbb Z)\le k_2\le4. $$ I'll be happy to upvote any answer improving the above estimate for $k_2(\mathbb Z)$!

Ewan Delanoy asked a related question.

For a list of the MSE questions about linear combination of idempotents via Google, click here, or (if it doesn't work) paste the words

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Pierre-Yves Gaillard
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1 Answers1

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The following answer is unfortunately very partial.

We have

(a) $k_2(\mathbb Z)\ge3$,

so that $k_2(\mathbb Z)$ is equal to $3$ or $4$ (and $k_2$ is also equal to $3$ or $4$).

For the sake of emphasis, let me repeat one of the questions:

Is any two by two integer matrix a $\mathbb Z$-linear combination of three idempotents?

To prove (a) it suffices to note that, even modulo $4$, the matrix $$ \begin{pmatrix}2&1\\0&2\end{pmatrix} $$ is not a $\mathbb Z$-linear combination of two idempotents. (The verification is straightforward.)

Pierre-Yves Gaillard
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