(a) Consider the quadratic form Q(x1, x2, x3) =(x² + x²) − 3x1x2 + 2(x1X3 + X2X3) = (X1, X2, X3) A Find the symmetric matrix A and apply the leading principal minor test to show that is indefinite. Answer: The symmetric matrix A is given by ( 1/2 -3/2 1 -3/2 1/2 1 1 1 0 (3) Xx2 A = = With LPM3 = det (A) = -4 <0, LPM₂ = -2 <0, LPM₁ = 1/2> 0 the LPM test tells us we have an indefinite quadratic form. (b) Compute the eigenvalues and the corresponding unit eigenvectors, and use them to construct the orthogonal matrix O diagonalising A. Give the normal form of and construct the variables in terms of which can be written as a sum of squares, i.e., determine variables ₁, i = 1,2,3, such that adopts the form Q = A₁²³ +₂2² + ¸Ã²³, and confirm from the eigenvalues that is indefinite.
(a) Consider the quadratic form Q(x1, x2, x3) =(x² + x²) − 3x1x2 + 2(x1X3 + X2X3) = (X1, X2, X3) A Find the symmetric matrix A and apply the leading principal minor test to show that is indefinite. Answer: The symmetric matrix A is given by ( 1/2 -3/2 1 -3/2 1/2 1 1 1 0 (3) Xx2 A = = With LPM3 = det (A) = -4 <0, LPM₂ = -2 <0, LPM₁ = 1/2> 0 the LPM test tells us we have an indefinite quadratic form. (b) Compute the eigenvalues and the corresponding unit eigenvectors, and use them to construct the orthogonal matrix O diagonalising A. Give the normal form of and construct the variables in terms of which can be written as a sum of squares, i.e., determine variables ₁, i = 1,2,3, such that adopts the form Q = A₁²³ +₂2² + ¸Ã²³, and confirm from the eigenvalues that is indefinite.
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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can you explain part 3b fully please
Transcribed Image Text:(a) Consider the quadratic form
Q(x1, x2, x3) = (x² + x²) − 3x1x2 + 2(x1X3 + X2X3) = (X1, X2, X3) A
Find the symmetric matrix A and apply the leading principal minor test to show
that is indefinite.
Answer: The symmetric matrix A is given by
(
1/2 -3/2 1
-3/2 1/2 1
1 1 0
(3)
A =
With LPM3 = det (A) = -4 < 0, LPM₂ = -2 <0, LPM₁ = 1/2 > 0 the LPM
test tells us we have an indefinite quadratic form.
(b) Compute the eigenvalues and the corresponding unit eigenvectors, and use them
to construct the orthogonal matrix O diagonalising A. Give the normal form of
and construct the variables in terms of which can be written as a sum of
squares, i.e., determine variables x₁, i= 1,2,3, such that adopts the form
Q = λ₁x² + √₂x² + √3x¹²,
and confirm from the eigenvalues that is indefinite.
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