Matrix - Inner Products and Dot Products

In summary, the conversation is about considering a matrix A in R^(nxn) and defining the product <x, y> = ((Ax)^t)(Ay) for vectors x and y in R^n. The questions posed are for which choices of A is this product an inner product and for which choices is it equal to the dot product. The equations for inner products are also mentioned. The solution given is that <x, y> = Ax . Ay and the difference between matrix multiplication and a matrix dot product is requested. Examples are also requested.
  • #1
essie52
10
0

Homework Statement



Consider a matrix A in R^(nxn) . In R^n for vectors x and y, define the product

<x, y> = ((Ax)^t)(Ay).

a) For which choices of A is this an inner product?

b) For which choices of A is <x, y> = x (dot) y (the dot product)?



Homework Equations



For inner products:

a. <x, y> = <y, x>
b. <x + y, z> = <xz> + <yz>
c. <cx, y> = c<x,y>
d. <x, x> is less than 0, for all nonzero x in A


The Attempt at a Solution


<x, y> = ((Ax)^T)(Ay) = Ax . Ay
 
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  • #2
So, what did you already try to solve the problem?
 
  • #3
Never mind. I figured it out but thanks anyway.
 
  • #4
whats the difference between matrix multiplication and a matrix dot product Give examples please
 

1. What is the difference between Inner Products and Dot Products?

The main difference between Inner Products and Dot Products is that Inner Products are defined for complex vector spaces while Dot Products are defined for real vector spaces. In addition, Inner Products are more general and can take on different forms, while Dot Products are always in the form of a sum of products.

2. How do you calculate the Inner Product of two vectors?

To calculate the Inner Product of two vectors, you first need to make sure they are in the same space. Then, you multiply the corresponding components of each vector and add them together. For example, if the vectors are in R^n, the Inner Product would be calculated as (a1*b1 + a2*b2 + ... + an*bn).

3. What is the purpose of the Inner Product in linear algebra?

The Inner Product is a fundamental concept in linear algebra that allows us to measure the similarity between vectors. It also allows us to define important concepts such as vector length, angle between vectors, and orthogonality. Additionally, Inner Products are used in a variety of applications, including in physics, engineering, and data analysis.

4. Can any two vectors have an Inner Product?

No, not all vectors can have an Inner Product. In order for two vectors to have an Inner Product, they must be in the same vector space and satisfy certain properties. For example, the vectors must be of the same dimension and the Inner Product must be commutative and satisfy the distributive property.

5. How does the Inner Product relate to the concept of orthogonality?

The Inner Product is closely related to the concept of orthogonality. Two vectors are considered orthogonal if their Inner Product is equal to 0. This means that the vectors are perpendicular to each other, and their angle is 90 degrees. Orthogonality is an important concept in linear algebra and is used in various applications, such as in geometric transformations and signal processing.

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