MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given an $n \times n$ matrix $A$ with entries in a commutative and associative ring with $1$ (say $Z[x_{1},\dots,x_{n^{2}}]$), the following paper guarantees existence of seven $B_{i}$s such that $A = \sum_{i=1}^{7}B_{i}^{k}$ with $k \le n$.

Is there an explicit algorithm to find the seven $B_{i}$ matrices?

share|cite|improve this question
The article you cite mention matrices with integer entries, which is a crucial hypothesis. With complex entries, I think 2 are enough. – Federico Poloni Dec 18 '11 at 14:42
@Federico: Thankyou, I am looking for integers for now. However where is the complex entries reference? – Turbo Dec 18 '11 at 16:53
I don't have a ready reference, just an idea for the proof (that's why I wrote "I think"). If a matrix has all distinct eigenvalues, then it is a $n$-th power (take $n$-th roots of the eigenvalues, keep the eigenvectors). So it is sufficient to find $C$ and $D$ with distinct eigenvalues such that $A=C+D$. We may assume without loss of generality that $A$ is in Jordan form, and then we can choose easily a diagonal $D$ such that $D$ and $C=A-D$ have distinct diagonal entries. – Federico Poloni Dec 18 '11 at 18:14
Is there a good algorithm for the "usual" Waring problem? The usual circle problem arguments seem to be purely enumerative, and not so much constructive... – Igor Rivin Dec 18 '11 at 21:44
@Igor Rivin: This one may be easier and that is probably why it was solved so quickly! – Turbo Dec 19 '11 at 0:56

The result mentioned above is Waring's problem for matrices (respectively for algebraic number fields). The result is:

Theorem (Katre, Kuhle 1990): Let $R$ be an order in an algebraic number field $K$. Let $n\ge k \ge 2$. Then every $n\times n$ matrix over $R$ is a sum of $k$-th powers if and only if it is the sum of seven $k$-th powers if and only if $(k, disc (R)) = 1$.

If this is true, then the question is if the seven matrices can be constructed explicitly. For the case $k=n=2$ and $R=\mathbb{Z}$ this seems to be the case (article of Newman "Sums of squares of matrices"), by constructing certain companion matrices to characteristic polynomials: every integral $2 \times 2$ matrix is the sum of at most $3$ integral squares. The proof for the general case however uses an argument of the form "every $n\times n$-matrix is a sum of $k$-th powers in $M_n(R)$ if every $m\times m$-matrix is a sum of $k$-th powers in $M_m(R)$ for $n\ge m\ge 1$ and $k\ge 2$", which does not seem to be constructive.

share|cite|improve this answer
The article clearly says $R$ is commutative and associative ring with $1$. Katre's works as I had inferred is more number-theoretic. – Turbo May 16 '13 at 5:53
Yes, but the results in Katre's work are often also for an arbitrary commutative and associative ring with unity, see And you asked yourself for the special case $R=\mathbb{Z}[x_1,\ldots ,x_m]$. It seems very interesting to consider also rings of integers etc. From the algorithmic point of view this should be better, too. – Dietrich Burde May 16 '13 at 9:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.