left-inverse and right-inverse

left-inverse and right-inverse

本科学习线性代数的时候,只讲过 逆 (inverse) 的概念。这里补充一些知识

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

证明: \[
B(AC) = (BA)C \quad \text {gives} \quad BI = IC \quad \text {which is} \quad B = C
\]
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.

因此我们得到结论:

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 \(A^{-1}\)

下面的结论很重要:

  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: \[
B = (A^T A)^{-1} A^T \\
C = A^T (A A^T)^{-1}
\]
显然, \(BA = I\) 同时 \(AC = I\)

唯一需要解决的问题是,如何确保 \((A^T A)\) 或者 \((A A^T)\) 存在逆矩阵 ?

这里需要利用矩阵 \(A^T A\) 的性质

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

如果 \(A\) 为方阵,即: If \(m = n\), we cannot have one property without the other. A square matrix has a left-inverse if and only if it has a right-inverse. There is only one inverse, namely \(B = C = A^{-1}\) .

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

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