Q68E If q is a quadratic form on Rn聽... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q68E (page 403)

If q is a quadratic form on $R^{n}$ with symmetric matrix A, and if is a linear transformation from $R^{m} to R^{n}$ show that the composite function $p (\overset{鈫�}{x}) = q (L (\overset{鈫�}{x}))$ is a quadratic form on role="math" localid="1659689309678" $R^{m}$ Express the symmetric matrix of p in terms of R and A.

Short Answer

Expert verified

Definition of functions and the quadratic form q is used to prove this problem and p is defined by the symmetric matrix $B = R^{T} AR$ .

Step by step solution

0f 2: Given information

Let us use the definitions of the functions p.L and the quadratic form q defined by the symmetric matrix A.

0f 2: Application

We have , $p (\overset{鈫�}{x}) = q (L (\overset{鈫�}{x})) = q (R \overset{鈫�}{x}) = (R \overset{鈫�}{x}) (A) (R \overset{鈫�}{x}) = ({\overset{鈫�}{x}}^{T} R^{T}) (A) (R \overset{鈫�}{x}) = {\overset{鈫�}{x}}^{T} (R^{T} AR) \overset{鈫�}{x} = {\overset{鈫�}{x}}^{T} B \overset{鈫�}{x}$
Here $B = R^{T} AR$ , is an symmetric matrix, where

$B^{T} = {(R^{T} AR)}^{T} = R^{T} A^{T} {(R^{T})}^{T} = R^{T} AR$

Therefore is a quadratic form in variables defined by the symmetric matrix $B = R^{T} AR$ .

Result:

$p (\overset{鈫�}{x})$ is a quadratic form which is Proved using definitions of the functions and the quadratic form q. p is defined by the symmetric matrix $B = R^{T} AR$ .

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91影视

If q is a quadratic form on Rnwith symmetric matrix A, and if is a linear transformation from RmtoRnshow that the composite function p(x鈫�)=q(Lx鈫�)is a quadratic form on role="math" localid="1659689309678" Rm Express the symmetric matrix of p in terms of R and A.

Short Answer

Step by step solution

0f 2: Given information

0f 2: Application

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.