9.4.1 Identity as Matrix Matrix Addition Example

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Identity Matrix as Matrix Matrix Addition

Let $I_{n} \in R^{n \times n}$ be the 8.6 Identity Matrix and set $n = 3$ .

I_{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

Claim. The identity matrix can be rewritten as a matrix sum using 9.4 Matrix-Matrix Addition.

I_{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]

Our matrix sum can then be rewritten again as an 9.1 Outer Product of Vectors.

\begin{aligned} I_{3} & = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}] [\begin{array}{c} 1 & 0 & 0 \end{array}] + [\begin{array}{c} 0 \\ 1 \\ 0 \end{array}] [\begin{array}{c} 0 & 1 & 0 \end{array}] + [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}] [\begin{array}{c} 0 & 0 & 1 \end{array}] \\ = {\vec{e}}_{1} \cdot {\vec{e}}_{1}^{T} + {\vec{e}}_{2} \cdot {\vec{e}}_{2}^{t} + {\vec{e}}_{3} \cdot {\vec{e}}_{3}^{T} \end{aligned}

We can rewrite this as a sum using sum notation.

I_{3} = \sum_{j = 1}^{3} {\vec{e}}_{j} \cdot \vec{e} j^{T}

Therefore, we can generalize and write $I_{n}$ as the sum

I_{n} = \sum_{j = 1}^{n} {\vec{e}}_{j} \cdot {\vec{e}}_{j}^{T}

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