Left inverse and right inverse

1. Overview

2. Description

2.1 Tall matrix and left inverse

Now the thing is that this matrix definitely does not have a full inverse. However, with a little bit of creativity, we can come up with a way to multiply this matrix to produce the identity matrix.

However, what we are inverting is the product A transpose A. So it turns out that A transpose A inverse times A transpose A is going to produce the identity matrix.

So it's three matrices three A's multiplied together and this is the left inverse.

that's the condition for a matrix, a tall matrix, having a left inverse so a tall matrix never has a full inverse but it can have or it does have a left inverse if it is full column rank.

The wide matrix does not have a left inverse.

2.2 Wide matrix and right inverse

If a is a wide matrix and is full row rank then it will have a right inverse. In other words, if the rank is M. In this video, I showed you how to compute the 2 one-sided inverses,

3. Reference