left-inverse and right-inverse

# left-inverse and right-inverse

The matrix $$A$$ cannot have two different inverses, Suppose $$BA = I$$ and also $$AC = I$$. Then $$B = C$$,

This shows that a left-inverse B (multiplying from the left) and a right-inverse C (multiplying $$A$$ from the right to give $$AC = I$$ ) must be the same matrix.

solving $$A A^{-1} = I$$ has at the same time solved $$A^{-1} A = I$$

=> A 1-sided inverse of a square matrix is automatically a 2-sided inverse

full rank: the rank = number of columns = number of rows

=> when the rank is as large as possible, $$r = n$$ or $$r = m$$ or $$r = m = n$$ , the matrix has a left-inverse B or a right-inverse C or a two-sided $$A^{-1}$$

1. EXISTENCE: Full row rank (行满秩) $$r = m$$.
$$Ax = b$$ has at least one solution for every $$b$$ if and only if the columns span $$\mathbb R^{m}$$. Then $$A$$ has a right-inverse such that $$AC = I_m$$ ( m by m ). This is possible only if $$m \le n$$

2. UNIQUENESS: Full column rank (列满秩) $$r = n$$.
$$Ax = b$$ has at most one solution $$x$$ for every $$b$$ if and only if the columns are linearly independent. Then $$A$$ has an $$n$$ by $$m$$ left-inverse B such that $$BA = I_n$$ (n by n). This is possible only if $$m \ge n$$

One-sided inverses: $B = (A^T A)^{-1} A^T \\ C = A^T (A A^T)^{-1}$

$$A^T A$$ have an inverse if the rank is $$n$$ and $$AA^T$$ has an inverse when the rank is $$m$$

Existence implies uniqueness and uniqueness implies existence, when the matrix is square.

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