9.2 Example of Matrix Units

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 9.1 Outer Product of Vectors

Properties: N/A
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Matrix Units

Let ${{\vec{e}}_{k}}_{k = 1}^{n} \in R^{n}$ , where $n$ is the amount of vectors, represent the elementary basis vectors for $R^{n}$ . We will define each vector ${\vec{e}}_{i}$ 's component as follows:

{\vec{e}}_{k} (j) = {\begin{cases} 1 & j = k \\ 0 & otherwise \end{cases}

For $n = 3$ , we see

\begin{aligned} {\vec{e}}_{1} = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}], & {\vec{e}}_{2} = [\begin{array}{c} 0 \\ 1 \\ 0 \end{array}], & {\vec{e}}_{3} = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}] . \end{aligned}

Notice how amount $n = 3$ and index $k$ of the vectors corresponds to the row with value $1$ .

Using these three vectors and the 9.1 Outer Product of Vectors, we can create $n \times n$ matrix units

E_{i k} = [{\vec{e}}_{i}]_{m \times 1} \cdot [{\vec{e}}_{k}^{T}]_{1 \times n}

with all zero entries except the entry in the row $i$ and column $k$ . Therefore, when $n = 3$ , we see the first matrix unit

\begin{aligned} E_{11} & = [{\vec{e}}_{1}]_{3 \times 1} \cdot [{\vec{e}}_{1}^{T}]_{1 \times 3} \\ = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}] [\begin{array}{c} 1 & 0 & 0 \end{array}] \\ = [\begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] \end{aligned}

For a $3 \times 3$ case, there are a total of nine matrix units. We can find these by repeating the process from above of multiplying ${\vec{e}}_{i}$ and ${\vec{e}}_{k}^{T}$ , AKA taking the 9.1 Outer Product of Vectors.

\begin{aligned} E_{11} = [\begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], & E_{12} = [\begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], & E_{13} = [\begin{array}{c} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], \\ E_{21} = [\begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}], & E_{22} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}], & E_{23} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}], \\ E_{31} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}], & E_{32} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array}], & E_{33} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}] . \end{aligned}