9.5.1 Matrix as Linear Combination of Outer Products

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Properties: 9.6 Algebraic Properties of Matrix Operations
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Matrix as Linear Combination of Outer Products

A = [\begin{matrix} 2 & - 1 \\ - 3 & 5 \end{matrix}]

Solution. We can start by splitting matrix $A$ into a matrix sum of four individual single-valued matrices.

\begin{aligned} [\begin{array}{c} 2 & - 1 \\ - 3 & 5 \end{array}] & = [\begin{array}{c} 2 & 0 \\ 0 & 0 \end{array}] + [\begin{array}{c} 0 & - 1 \\ 0 & 0 \end{array}] + [\begin{array}{c} 0 & 0 \\ - 3 & 0 \end{array}] + [\begin{array}{c} 0 & 0 \\ 0 & 5 \end{array}] \end{aligned}

To create a linear combination, we simply factor out the corresponding entries of each matrix.

= 2 \cdot [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] + - 1 \cdot [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] + - 3 \cdot [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] + 5 \cdot [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]

Finally, we can rewrite the matrices as matrix units using 9.1 Outer Product of Vectors.

= 2 \cdot {\vec{e}}_{1} \cdot {\vec{e}}_{1}^{T} + - 1 \cdot {\vec{e}}_{2} \cdot {\vec{e}}_{2}^{T} + - 3 \cdot {\vec{e}}_{3} \cdot {\vec{e}}_{3}^{T} + 5 \cdot {\vec{e}}_{4} \cdot {\vec{e}}_{4}^{T}