All row exchange matrices are symmetric: ${P}^{T} = P$ . Then ${P}^{T} P = I$ becomes ${P}^{2} = I$ . Other permutation matrices mayor may not be symmetric.
(a) If P sends row 1 to row 4, then ${P}^{T}$ sends row ____ to row ____. When ${P}^{T} = P$ the row exchanges come in pairs with no overlap.
(b) Find a 4 by 4 example with ${P}^{T} = P$ that moves all four rows.

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Accepted Answer

(a) If P sends row 1 to row 4, then [latex]{P}^{T}[/latex] sends row 4 to row 1

(b) [latex]P=\begin{bmatrix} E & 0 \ 0 & E \end{bmatrix}={P}^{T}[/latex] with [latex]E=\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}[/latex] moves all rows:

1 and 2 are exchanged, 3 and 4 are exchanged.

Question 2.7.15: All row exchange matrices are symmetric: P^T = P. Then P^T P...

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