(a) Define the quadratic form qA(x1, · · · , xn) associated to a real symmetric n × nmatrix A. (b) Show that if two real symmetric matrices A, B are congruent then there is alinear 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 nmatrix A.(b) Show that if two real symmetric matrices A, B are congruent then there is alinear 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 byV₁ • 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 productspace with the stated dot products?(Hint: you may wish to diagonalise the associated quadratic formq(x, y) = x² + 2xy + 2y².)

“Since you have posted multiple questions, we will provide the solution only to the first question…

Answered: (a) Define the quadratic form qA(x1, ·…

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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.)