cancel
Showing results for 
Search instead for 
Did you mean: 
cancel
Showing results for 
Search instead for 
Did you mean: 

Community Tip - Your Friends List is a way to easily have access to the community members that you interact with the most! X

Non-square matrix with left and right inverse matrices

ValeryOchkov
24-Ruby IV

Non-square matrix with left and right inverse matrices

A non-square matrix can have left and right inverse matrices.

Left and right inverse matrices may not be the only ones. If a matrix has both left and right inverse, then it is invertible and m = n.
Let M be an m x n matrix, i.e. defining a linear mapping from Rn to Rm.
1. If n < m, then there can exist a left inverse L matrix n x m from Rm to Rn such that LA = E (the identity matrix), the condition of this: Ker A = 0 (the mapping A is one-to-one).

LIM.png
2. If n > m, then there can exist a right inverse R n x m matrix from Rm to Rn such that AR = E, the condition of this: Im A = Rn (mapping of A to the whole space).
RIM.png

How can we calculate RIM?

 

 

ACCEPTED SOLUTION

Accepted Solutions

 


@ValeryOchkov wrote:

A non-square matrix can have left and right inverse matrices.

Left and right inverse matrices may not be the only ones. If a matrix has both left and right inverse, then it is invertible and m = n.
Let M be an m x n matrix, i.e. defining a linear mapping from Rn to Rm.
2. If n > m, then there can exist a right inverse R n x m matrix from Rm to Rn such that AR = E, the condition of this: Im A = Rn (mapping of A to the whole space).
RIM.png

How can we calculate RIM?

 

 


?

 

2021 11 02 A.png

 

Stuart

View solution in original post

3 REPLIES 3

 


@ValeryOchkov wrote:

A non-square matrix can have left and right inverse matrices.

Left and right inverse matrices may not be the only ones. If a matrix has both left and right inverse, then it is invertible and m = n.
Let M be an m x n matrix, i.e. defining a linear mapping from Rn to Rm.
2. If n > m, then there can exist a right inverse R n x m matrix from Rm to Rn such that AR = E, the condition of this: Im A = Rn (mapping of A to the whole space).
RIM.png

How can we calculate RIM?

 

 


?

 

2021 11 02 A.png

 

Stuart

Personally I would prefer Stuarts way, but if you feel so, you may also use "geninv" to get the desired result:

Werner_E_0-1635870021334.png

 

The calculation which is done behind the scenes when using the above shown way with "geninv" is actually this:

Werner_E_1-1635870174402.png

 

new-geninv.png

Announcements

Top Tags