- #1

PainterGuy

- 940

- 70

- Homework Statement
- I was working with right inverse and left inverse of a matrix.

- Relevant Equations
- Please check my work below.

Hi,

It's actually not a homework problem but I still decided to post it here.

Problem:

Consider Ax=y, where A is mxn and has rank m. Is (A′A)⁻¹A′y a solution? If not, under what condition will it be a solution? Is A′(AA′)⁻¹y a solution?

The given solution is:

Consider Ax=y with A mxn and rank(A)=m, which implies n⋝m. If n>m then A′A is nxn and singular . Thus (A′A)⁻¹ is not defined and (A′A)⁻¹A′y is not a solution. Because AA′ is mxm and nonsingular, substituting A′(AA′)⁻¹y into Ax yields y. Thus A′(AA′)⁻¹y is a solution. If n=m then both reduce to A⁻¹y and are solutions of Ax=y.

A′(AA′)⁻¹ is right inverse but the system in question statement is written as "Ax=y". The yellow highlighted text below says that a right inverse, A′(AA′)⁻¹, is useful for solving XA=Y system. You notice the contradiction. Which source is correct? Given solution or the yellow highlighted text? Could you please help me?

Source: https://math.stackexchange.com/a/1335707/775285

Mostly it's written AX=B where B is a constant matrix. In the case above the system is written instead as Ax=y. Is B=y? This link is relevant here . Could you please help me with it?

It's actually not a homework problem but I still decided to post it here.

Problem:

Consider Ax=y, where A is mxn and has rank m. Is (A′A)⁻¹A′y a solution? If not, under what condition will it be a solution? Is A′(AA′)⁻¹y a solution?

The given solution is:

Consider Ax=y with A mxn and rank(A)=m, which implies n⋝m. If n>m then A′A is nxn and singular . Thus (A′A)⁻¹ is not defined and (A′A)⁻¹A′y is not a solution. Because AA′ is mxm and nonsingular, substituting A′(AA′)⁻¹y into Ax yields y. Thus A′(AA′)⁻¹y is a solution. If n=m then both reduce to A⁻¹y and are solutions of Ax=y.

**Question 1:**A′(AA′)⁻¹ is right inverse but the system in question statement is written as "Ax=y". The yellow highlighted text below says that a right inverse, A′(AA′)⁻¹, is useful for solving XA=Y system. You notice the contradiction. Which source is correct? Given solution or the yellow highlighted text? Could you please help me?

Source: https://math.stackexchange.com/a/1335707/775285

**Question 2:**Mostly it's written AX=B where B is a constant matrix. In the case above the system is written instead as Ax=y. Is B=y? This link is relevant here . Could you please help me with it?