9.3 Example of Outer Product

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Using Outer Product to Generate Matrix

Let's generate the matrix using an 9.1 Outer Product of Vectors.

T = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \end{matrix}]}_{5 \times 5}

Solution. To generate the $5 \times 5$ matrix using an outer product, we first need to find the $1 \times 5$ and $5 \times 1$ vectors that we can multiply to give us our values in their corresponding spots within the matrix $T_{2}$ .

\begin{aligned} T_{32} & = 3 \\ T_{42} & = - 2 \\ T_{52} & = 4 \end{aligned}

Notice how all these values are in column $2$ . We can organize the column $2$ values into a $5 \times 1$ 3.1 Column Vector

\vec{c} = [\begin{matrix} 0 \\ 0 \\ 3 \\ - 2 \\ 4 \end{matrix}]

To create a $5 \times 5$ matrix from this, we need to multiply $\vec{c}$ with the 4.4 Transpose of a Vector.

\begin{aligned} \vec{r} & = [\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}] \\ ⟹ {\vec{r}}^{T} & = [\begin{array}{c} 0 & 1 & 0 & 0 & 0 \end{array}] \end{aligned}

The second value of the vector, ${\vec{r}}_{2}$ is $1$ because we want to place our values in column $2$ of our matrix.
By the identity property of multiplication, we know $1 \times n$ is $n$ .

We simply multiply the column vector $\vec{c}$ and the row vector ${\vec{r}}^{T}$ , also known as taking the 9.1 Outer Product of Vectors.

\begin{aligned} \vec{c} \cdot {\vec{r}}^{T} & = [\begin{array}{c} 0 \\ 0 \\ 3 \\ - 2 \\ 4 \end{array}] [\begin{array}{c} 0 & 1 & 0 & 0 & 0 \end{array}] \\ = {[\begin{array}{c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \end{array}]}_{5 \times 5} \end{aligned}

Our final matrix is equal to $T$ .

Why do we use column vectors?

The reason we use two column vectors with one of them transposed instead of just being given a row vector is because paper is longer than it is wide. It is much easier to write downwards than across the page. Column vectors are also easier to read because we read English from left to right.