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## Non-square matrix with left and right inverse matrices  24-Ruby III

## 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). 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). How can we calculate RIM?

1 ACCEPTED SOLUTION

Accepted Solutions  23-Emerald II
(To:ValeryOchkov)

@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). How can we calculate RIM?

? Stuart

3 REPLIES 3  23-Emerald II
(To:ValeryOchkov)

@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). How can we calculate RIM?

? Stuart  24-Ruby V
(To:ValeryOchkov)

Personally I would prefer Stuarts way, but if you feel so, you may also use "geninv" to get the desired result: The calculation which is done behind the scenes when using the above shown way with "geninv" is actually this:   24-Ruby III
(To:Werner_E)  Announcements