Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

Based on the tags, I'm assuming that the $x_i$ and $a_i$ are supposed to be integers. The procedure below generalizes to the case where you want simultaneous solutions to multiple equations. (It also works the same way if we use any principal ideal domain in place of integers.)

Write the inputs as a $1 \times n$ matrix $X = \begin{pmatrix}x_1 & \dotsb & x_n\end{pmatrix}$; assume $X \ne 0$ or else the problem is trivial. The problem is to find the vectors $A = \begin{pmatrix}a_1 & \dotsb & a_n\end{pmatrix}^T$ such that $XA = 0$. Smith normal form says that we can find an invertible $n \times n$ matrix $T$ such that $XT = \begin{pmatrix}d & 0 & \dotsb & 0\end{pmatrix}$, where $d = \gcd(x_1, \dotsc, x_n)$. (There would also be a $1 \times 1$ matrix $S$ that goes on the left, but we can just take it to be $1$.) (from Procedures to find solution to $a_1x_1+\cdots+a_nx_n = 0$)

So, what happens if $X$ contains zero entry - but is not zero vector?

share|cite|improve this question

Your Answer


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

Browse other questions tagged or ask your own question.