置换矩阵的逆等于其转置

Permutations

Definition

A permutation matrix \(P\) has the rows in \(I\) in any order.

Properties

\[P^{-1}=P^T \quad or \quad PP^T=I \]

Proof:

Suppose \(A=PP^T\), the columns of \(A\) are combinations of columns of \(P\). Since \(P^T\) is also permutation matrix, the columns of \(A\) are actually an arrangement of columns of \(P\). And \(\forall j\), the \(j\)'s column of A is actually determined by the row of \(P^T\) that number "\(1\)" appears in the \(j\)'s column.

The \(j\)'s column of \(A\) is actually the \(i\)'s column of \(P\), where \((P^T)_{ij}=1\). Since \((P^T)_{ij}=P_{ji}\), \(\forall (P^T)_{ij}=1,P_{ji}=1\). As a result, \(A_{jj}=1, \forall j\), and which means \(A=I\).

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