Matrix - Inner Products and Dot Products

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Homework Help Overview

The discussion revolves around the properties of inner products defined by a matrix A in R^(nxn) and their relationship to the standard dot product in R^n. Participants explore the conditions under which the defined product = ((Ax)^t)(Ay) qualifies as an inner product and when it aligns with the traditional dot product.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants consider the properties required for to be an inner product, referencing conditions such as symmetry and linearity. There is also a question about the distinction between matrix multiplication and the dot product, with requests for examples.

Discussion Status

The discussion includes attempts to clarify definitions and properties related to inner products. One participant indicates they have resolved their query independently, while others are still seeking clarification on related concepts.

Contextual Notes

Participants are navigating the definitions and properties of inner products and dot products, with some expressing uncertainty about the implications of matrix operations in this context.

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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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So, what did you already try to solve the problem?
 
Never mind. I figured it out but thanks anyway.
 
whats the difference between matrix multiplication and a matrix dot product Give examples please
 

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