Give a matrix $A$ whose size e.g., $m$ x $n$, and $m<n$. I know the the null space of that matrix is $X$ such that $A*X = 0$. The nullity of matrix $X$ is e.g $r$. it means the null space I got is a matrix of size $m$ x $r$. As shown HERE, I can build the square $m$ x $m$ matrix $Y$ by taking the values of each column from the null space such that $A*Y = 0$.

My question, what is the maximum rank of $Y$ we can get by taking its value from the null space? Is it always equal to the nullity $r$? or it can be bigger?


1 Answer 1


The rank of a matrix $M$ is by definition the dimension of the vector space generated by its columns. If the columns belong to a space of dimension $r$, they generate a space of dimension $\leq r$, whence $rank(M)\leq r$.

  • $\begingroup$ you mean maximum rank of matrix $Y$ in my case must be $r$ since I built $Y$ based on a matrix $X$ whose rank is $r$? $\endgroup$
    – Fatima_Ali
    May 20, 2021 at 10:43
  • $\begingroup$ @Fatima_Ali Exactly. The maximum rank of $Y$ is the dimension of the null-space of $A$. $\endgroup$
    – Desperado
    May 20, 2021 at 10:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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