(a) Define the quadratic form qÃ(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 x₁,...,xn such that ¶₁(x₁, ,xn) = 9B (x₁, ,xn). (c) Show that the following quadratic form on R³ is not positive definite: q(x, y, z) = x² + 4xz + 3y² + 4yz + z². ... (d) Let V be a real vector space with basis v₁, v2 and define a dot product by V₁ V₁ = 1, V₁ V2 V₂2 V₁ = λ₁ . V₂ V2 = 2, where A E R is fixed. For what values of A 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 + 2xXy + 2y².)
(a) Define the quadratic form qÃ(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 x₁,...,xn such that ¶₁(x₁, ,xn) = 9B (x₁, ,xn). (c) Show that the following quadratic form on R³ is not positive definite: q(x, y, z) = x² + 4xz + 3y² + 4yz + z². ... (d) Let V be a real vector space with basis v₁, v2 and define a dot product by V₁ V₁ = 1, V₁ V2 V₂2 V₁ = λ₁ . V₂ V2 = 2, where A E R is fixed. For what values of A 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 + 2xXy + 2y².)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:(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 x₁,, such that qA (X₁, ···,xn) = ¶B(x₁, ···, x').
9
(c) Show that the following quadratic form on R³ is not positive definite:
q(x, y, z) = x² + 4xz + 3y² + 4yz + z².
x
n
(d) Let V be a real vector space with basis V₁, V2 and define a dot product by
V1 V1 1,
V2 V1 = 1, V2 V2 =
=
V1 V2
=
●
●
2,
where A E R is fixed. For what values of X 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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Follow-up Question
Let V be a real
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? [7]
(Hint: you may wish to diagonalise the associated quadratic form
q(x, y) = x
2 + 2xλy + 2y
2
.)
Solution
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