right and left inverse of a matrix

[DrDolittle]

This is a condensation of my understanding of existence of matrix inverses.

We know that for a matrix to be invertible, its rank should be as large as possible. Let m,n denote the order of the matrix A,that is, m*n, r denote the rank of the matrix.

For a square matrix r=m=n, and A*inverse(A) = inverse(A)*A = 1

For a rectangular matrix [ A*inverse(A) is not equal to inverse(A)*A]
1)if m>n
r=n,
No free variables or only pivot columns
No nullspace(only the zero vector)
atmost one solution

2)if m<n
r=m,
free variables exist
null space exists
atleast one solution

The only thing i didnt understand is how to deduce there exist a left inverse for the first condition and right inverse for the second condition.
Thanx in advance
Regards
drdolittle

 

[claudiocamera]

As far as I am concerned inverse matrix only exists if the matrix is square and when the rank is m if the matrix is mxm otherwise the matrix is singular.

What you are probably studying is the solution of linear systems whereas I have never seen this concept of rectangular inverse .

 

[DrDolittle]

nope the inverse exists

 

[mmatica]

You probably want the so called pseudoinverse. See:

https://en.wikipedia.org/wiki/Pseudoinverse

Mate

 

[alihzubi]

An m*n matrix has at least one left inverse iff it is injective, and at least one right inverse iff it is surjective. The infinitely many inverses come due to the kernels (left and right) of the matrix. If the matrix has no left nor right kernels; i.e.: it is square full rank matrix, the inverses collapse to unique inverse; the usual one.