9.10 Transposition Matrix

#Type/Definition #Type/Example #Topic/Linear_Algebra

Types: N/A
Examples: 9.10.1 Transposition Matrix as Outer Product
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
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Transposition Matrix

Let natural number $n \in N$ . For indices $i \neq k$ , the transposition matrix in $R^{n \times n}$ is given by

P_{i k} = \underset{j \neq i, k}{\sum_{j = 1}^{n}} {\vec{e}}_{j} \cdot {\vec{e}}_{j}^{T} + {\vec{e}}_{i} \cdot {\vec{e}}_{k}^{T} + {\vec{e}}_{k} \cdot {\vec{e}}_{i}^{T}

with all ${\vec{e}}_{i} \in R^{n}$ for all $i = 1, 2, \dots, n$ .

Remark. Notice how any transposition matrix can be formed by taking the square $n \times n$ 8.6 Identity Matrix and swapping row $i$ with row $k$ . Likewise, this can also be done by swapping columns because the identity matrix is symmetric. Transposition matrices are an example of rank-two updates because there are two linearly dependent vectors.
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Transposition Matrix

Let's consider the transposition matrix $P_{23}$ in the $n = 4$ case.

We can compute the shear matrix using our definition from above.

\begin{aligned} P_{23} & = P_{i k} \\ = \underset{j \neq i, k}{\sum_{j = 1}^{n}} {\vec{e}}_{j} \cdot {\vec{e}}_{j}^{T} + {\vec{e}}_{i} \cdot {\vec{e}}_{k}^{T} + {\vec{e}}_{k} \cdot {\vec{e}}_{i}^{T} & Transposition matrix definition \\ = \underset{j \neq 2, 3}{\sum_{j = 1}^{4}} {\vec{e}}_{j} \cdot {\vec{e}}_{j}^{T} + {\vec{e}}_{2} \cdot {\vec{e}}_{3}^{T} + {\vec{e}}_{3} \cdot {\vec{e}}_{2}^{T} & Plug in values \\ = \underset{sum constraint: j \neq 2, 3}{\underset{⏟}{{\vec{e}}_{1} \cdot {\vec{e}}_{1}^{T} + {\vec{e}}_{4} \cdot {\vec{e}}_{4}^{T}}} + {\vec{e}}_{2} \cdot {\vec{e}}_{3}^{T} + {\vec{e}}_{3} \cdot {\vec{e}}_{2}^{T} & Expand sum \\ = [\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array}] [\begin{array}{c} 1 & 0 & 0 & 0 \end{array}] + [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}] [\begin{array}{c} 0 & 0 & 0 & 1 \end{array}] + [\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}] [\begin{array}{c} 0 & 0 & 1 & 0 \end{array}] + [\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}] [\begin{array}{c} 0 & 1 & 0 & 0 \end{array}] & Evaluate \\ = [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}] + [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}] + [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}] + [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}] & Simplify \\ = \underset{identity matrix with i and k cancelled out}{\underset{⏟}{[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}]}} + \underset{i and k rows/columns swapped}{\underset{⏟}{[\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]}} \\ = [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}] & Final transposition matrix \end{aligned}

Remark. These types of matrices are part of a bigger group of matrices known as permutation matrices.
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