(SOLVED) Outer Product
Discipline: Mathematics
Type of Paper: Question-Answer
Academic Level: Undergrad. (yrs 3-4)
Paper Format: APA
Pages: 1
Words: 275
Question
The tensor product of two coordinate vectors is termed as “Outer
product”. This is a special case for “Kronecker product of matrices”.
Let u and v be vectors. Then, the outer product of u and v is w=uvT. The outer product is same as the matrix multiplication uvT also u is denoted by m × 1 column vector and v is denoted by n × 1 column vector.
Let
![](https://media.cheggcdn.com/study/494/494139cc-79a5-4578-a876-08d0cc07f2c9/DC-1776V2.png)
![](https://media.cheggcdn.com/study/8a3/8a3993b3-d3e3-4885-a922-78c73a0f1d76/DC-1776V3.png)
Properties of an outer product:
• The result of an outer product is m × n rectangular matrix.
• The outer product is not commutative. That is,
![](https://media.cheggcdn.com/study/654/6541b4ba-f786-4aec-a817-8ec900360d5b/DC-1776V4.png)
• Multiply the second vector v with the resultant product
![](https://media.cheggcdn.com/study/868/868b45ba-74c8-4516-8c5e-ccfc5977c0c3/DC-1776V5.png)
![](https://media.cheggcdn.com/study/67c/67c1c1ef-5069-4bc3-b000-ce383334adc3/DC-1776V6.png)
Example:
Consider the vectors
![](https://media.cheggcdn.com/study/b81/b818ce36-0bb5-49b6-9369-9f34509f46e8/DC-1776V7.png)
Transpose of v is, vT = [7 2 3 1].
The outer product of
![](https://media.cheggcdn.com/study/b71/b71bd389-4b40-42e9-9b9f-ba91163d6a7f/DC-1776V8.png)
![](https://media.cheggcdn.com/study/edd/edd9eb63-43c6-468a-8a9d-7f859ec21c79/DC-1776V9.png)
Thus, the outer product is a rectangular 3 × 4 matrix.
Check the outer product is commutative or not.
![](https://media.cheggcdn.com/study/7cd/7cdddafd-c3fd-4912-a5ec-d22ec6e0f4e8/DC-1776V10.png)