0
$\begingroup$

Suppose we have $A^{m*n}$ and $B^{n*m} $ matrices. What constraints has to be satisfied by $A$ and $B$ so that the product of these result in the identity matrix $I$. I fund that it is possible but I cannot find the constraints for $A$ and $B$ which I really need. Thanks for the help.

$\endgroup$
  • $\begingroup$ The constraints should, probably, be black magic or something like that: the dot or inner product of two vectors in a linear space with inner product gives us a scalar, not another vector. Perhaps you meant simply product? $\endgroup$ – DonAntonio May 16 '17 at 9:47
  • $\begingroup$ Yes sorry you are right $\endgroup$ – DalekSupreme May 16 '17 at 9:50
2
$\begingroup$

I suppose by dot product you mean usual matrix product.

So if $$AB=I_m$$ then $B$ is called the right inverse of $A$ and $A$ is the left inverse of $B$.

If $$BA=I_n$$ then $A$ is the right inverse of $B$ and $B$ is the left inverse of $A$.

A matrix has right inverse iff its columns are independent.

A matrix has left inverse iff its rows are independent.

However, a non-square cannot have both left and right inverse.

See: Inverse of non-square matrix

$\endgroup$
  • $\begingroup$ Thanks! Just what I needed :) $\endgroup$ – DalekSupreme May 16 '17 at 9:55
1
$\begingroup$

Too big for a comment, but for a pair of $2\times 3$ and $3\times 2$ matrices we get $$\left(\begin{array}{lll}a&b&c \\ d&e&f \end{array}\right)\left(\begin{array}{ll}g&h\\i&j\\k&l\end{array}\right)=\left(\begin{array}{ll}1&0\\0&1\end{array}\right).$$ This implies $$ag+bi+ck=1\\dg+ei+fk=0\\ah+bj+cl=0\\dh+ej+fl=1.$$ So these are the constraints, which you can adapt to any sized matrices.

$\endgroup$

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.