Linear algebraConcept reference

Outer products

The outer product of column vectors u and v is the matrix uv transpose, with entry (i,j) equal to u_i times v_j.

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01Understand the idea#

For column vectors $u$ and $v$ , the outer product $uv^T$ pairs every entry of $u$ with every entry of $v$ .

Let $u=(2,-1)$ and $v=(3,0,4)$ . Row 1 is 2 times $v^T$ , giving $(6,0,8)$ . Row 2 is $-1$ times $v^T$ , giving $(-3,0,-4)$ .

These two rows form

uv^T=\begin{pmatrix}6&0&8\\-3&0&-4\end{pmatrix}

If $u$ has $m$ entries and $v$ has $n$ , then $uv^T$ has $m$ rows and $n$ columns.

At row i and column j,

(uv^T)_{ij}=u_i v_j

Each cell contains one product. The two vector lengths may differ.

Check your reasoning

Column vectors $u=(-2,3)$ and $v=(4,-1)$ form $uv^T$ . Find entry $(2,1)$ .

Show answer and explanation

Row 2 uses 3; column 1 uses 4. Their product is 12.

Column times row gives $uv^T$ , a matrix. With equal lengths, row times column gives the scalar $u^Tv$ .

In $uv^T$ , a zero entry of $u$ makes a whole zero row. A zero entry of $v$ makes a whole zero column.

Check your reasoning

Batches have 2 and 5 kits. Each kit uses 3 pins and 4 clips. Choose row 2 (pins, clips).

Show answer and explanation

$(15,20)$

Five kits need 15 pins and 20 clips.

Check your reasoning

$u$ has 2 entries; $v$ has 3. Both are columns. Repair the claimed $3\times2$ shape of $uv^T$ .

Show answer and explanation

$2\times3$

Two rows from $u$ ; three columns from $v$ .

Key takeaway