(a) Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × n matrix A.
(a) Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × n matrix A.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 30E
Related questions
Question
-
(a) Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × n
matrix A. -
(b) Show that if two real
symmetric matrices A, B are congruent then there is a
linear change of variables to x′1,··· ,x′n such that qA(x1,··· ,xn) = qB(x′1,··· ,x′n). -
(c) Show that the following quadratic form on R3 is not positive definite:
q(x,y,z)=x^2 +4xz+3y^2 +4yz+z^2. -
(d) Let V be a real
vector space with basis v1,v2 and define a dot product by v1 ·v1 =1, v1 ·v2 =v2 ·v1 =λ, v2 ·v2 =2,where λ ∈ R is fixed. For what values of λ does (V, ·) become an inner product space with the stated dot products?
5. (Hint: you may wish to diagonalise the associated quadratic form q(x,y)=x^2 +2xλy+2y^2.)
![(a) Define the quadratic form qA(₁,,n) associated to a real symmetric n x n
matrix A.
(b) Show that if two real symmetric matrices A, B are congruent then there is a
linear change of variables to x₁,, such that q₁(x₁,,xn) = 9B(x₁, ···, x₂).
(c) Show that the following quadratic form on R³ is not positive definite:
q(x, y, z) = x² + 4xz + 3y² + 4yz + z².
n
(d) Let V be a real vector space with basis V₁, V2 and define a dot product by
V₁ • V₁ = 1, V1 · V2 = V2 · V1 = λ, V2 V2 =
2,
where À E R is fixed. For what values of λ does (V,.) become an inner product
space with the stated dot products?
(Hint: you may wish to diagonalise the associated quadratic form
q(x, y) = x² + 2xy + 2y².)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16556e8d-f4c5-4c0e-af47-6e925b310d10%2F004fe051-c03b-4a15-afb3-b24bcceb8034%2Ftn3tj4e_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Define the quadratic form qA(₁,,n) associated to a real symmetric n x n
matrix A.
(b) Show that if two real symmetric matrices A, B are congruent then there is a
linear change of variables to x₁,, such that q₁(x₁,,xn) = 9B(x₁, ···, x₂).
(c) Show that the following quadratic form on R³ is not positive definite:
q(x, y, z) = x² + 4xz + 3y² + 4yz + z².
n
(d) Let V be a real vector space with basis V₁, V2 and define a dot product by
V₁ • V₁ = 1, V1 · V2 = V2 · V1 = λ, V2 V2 =
2,
where À E R is fixed. For what values of λ does (V,.) become an inner product
space with the stated dot products?
(Hint: you may wish to diagonalise the associated quadratic form
q(x, y) = x² + 2xy + 2y².)
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