3
$\begingroup$

Let $u = \begin{bmatrix}a\\b\\c\end{bmatrix}$

Let $v = \begin{bmatrix}d\\e\\f\end{bmatrix}$

There exists a $3\times3$ matrix, $M$, such that:

$Mu = v$

so $M = vu^{-1}$

But how do I go about calculating $vu^{-1}$?

My guess is that $u^{-1}$ will be a row vector in order to make the multiplication work... But apparently you can't do inverses on non-square matrices.

$\endgroup$
5
$\begingroup$

NO!!! Vectors do not have inverses. There are infinitely many matrices $M$ such that $M u = v$ (as long as $u \ne 0$).

You could, for example, take $M = v w$ where $w$ is any row vector such that $w u = 1$. And you could add to $M$ any matrix $N$ such that $N u = 0$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ hmm but is there a way to create a general form for the matrix which satisfies such equation $\endgroup$ – Theo Walton Dec 17 '17 at 4:32
  • $\begingroup$ As I said, take one and add $N$ such that $N u = 0$. $\endgroup$ – Robert Israel Dec 17 '17 at 4:33
3
$\begingroup$

Suppose that you are given $u, v$ and want to find a particular $M$ that satisfy $Mu=v$, we can't talk about $u^{-1}$ as it is not even a square a matrix, it is not well defined.

  • if $u=0$ and $v=0$, $M$ can be any $3 \times 3$ matrix.
  • if $u=0$ and $v \neq 0$, no such $M$ exists.
  • if $u \neq 0$,suppose $u_i \neq 0$, let the $i$-th column of $M$ be $\frac{v}{u_i}$ and the other columns be zero columns.
| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

$$M u=v$$ is a linear system of equations in the unknowns $M_{ij}$. Like for any underdetermined linear system, you can find all its solutions using the pseudoinverse.

| cite | improve this answer | |
$\endgroup$

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.