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

Assume I have linear equation system in $n$ variables and the solution is $x_1=x_2=\cdots = x_n$.

What properties must the matrix of the system have, or how do I prove that the system has such a solution?

share|cite|improve this question
up vote 6 down vote accepted

I assume that you have an $n\times n$ linear system $Ax=0$, whose solutions are precisely $x_1=x_2=\cdots=x_n$. This means that the null space is the linear span of the vector $e=(1,1,\cdots,1)$. So the row space is the orthogonal complement of $e$. That means the sum of each row of $A$ must be 0, and that $n-1$ of these rows must be linearly independent.

share|cite|improve this answer

No. You can solve these type of equations irrespective of A being a square matrix or even otherwise. In Ax = 0, consider that A is a general mxn matrix. If m>>n, then you have a redundant system of equations. You can solve these equations using either of normal equations(pseudoinverse), Singular value decomposition or QR decomposition. Look at it as a least squares problem. x is a value which minimizes the difference between b(here it is zero) and Ax.

share|cite|improve this answer

Your Answer


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