Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a certain problem that I've managed to convert to a matrix problem. I have 3 unknown variables and the problem is defined by a 3x3 matrtix and a 3x1 vector. From the nature of the problem the rank is 2. My question is if it is possible to get the relation x/y - the first two unknowns. The specific problem is as follows: $$ \underbrace{\begin{pmatrix}0&2&1\\2&0&1\\1&3&2\end{pmatrix}}_{H} \ \underbrace{\begin{pmatrix}x\\y\\z\end{pmatrix}}_{t} =\underbrace{\begin{pmatrix}w_1\\w_2\\w_3\end{pmatrix}}_{b} $$ $\operatorname{rank}(H) = 2,\ \det(H) = 0$, $w_1,w_2,w_3$ are known scalars

thanks,

share|improve this question
add comment

2 Answers 2

up vote 2 down vote accepted

In general, I'm afraid not. You see, we have $$ H\begin{pmatrix}1\\1\\-1\end{pmatrix}=0. $$ Therefore, if $(x_0,y_0,z_0)$ is a solution to $Ht=b$, then $(x,y,z)=(x_0+k,y_0+k,z_0-k)$ is also a solution for any $k$. But the ratio $\frac{x}{y}=\frac{x_0+k}{y_0+k}$ is non-constant in general. The only exceptional case is $x_0=y_0=0$ and $(w_1,w_2,w_3)\not=0$, i.e. when $(w_1,w_2,w_3)$ is a nonzero multiple of $(1,1,2)$. Then we have $\frac xy=1$ for each nonzero solution $t$.

share|improve this answer
add comment

$Z$ can be a free variable (since $rank(H)=2$), from the first equation you can express $Y$ in terms of $Z$ and from the second equation you can express $X$ in terms of $Z$.

Now you have a general expression for $X/Y$ for a given a $Z$ - check if this ration is dependent on $z$, maybe you would like to set a $z$ so get a specific ratio if it is dependent on it.

share|improve this answer
add comment

Your Answer

 
discard

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.