Find the symmetric matrix A associated with the quadratic form Q(x1, x2, x3) = 2x1x2 + 2x1x3 + 2x2x3. Use the Principal Minor Test to establish whether Q is positive/negative (semi-)definite, or indefinite. Find the eigenvalues -1,-1,2 of the above quadratic forn Q and classify Q based on these eigenvalues. Are the two classifications, via the PMT and the eigenvalues, consistent? Find the corresponding unit eigenvectors. There will be two eigenvectors for the repeated eigen value A1,2 = -1; show that one can choose these such that the columns of O are othogo- nal. Construct the orthogonal matrix O from the three orthogonal unit eigenvectors. Prove that 0-1 OT. Provide the relation between x and x and confirm that Q(1, 2, 3) Q(x1, x2, x3). = =
Find the symmetric matrix A associated with the quadratic form Q(x1, x2, x3) = 2x1x2 + 2x1x3 + 2x2x3. Use the Principal Minor Test to establish whether Q is positive/negative (semi-)definite, or indefinite. Find the eigenvalues -1,-1,2 of the above quadratic forn Q and classify Q based on these eigenvalues. Are the two classifications, via the PMT and the eigenvalues, consistent? Find the corresponding unit eigenvectors. There will be two eigenvectors for the repeated eigen value A1,2 = -1; show that one can choose these such that the columns of O are othogo- nal. Construct the orthogonal matrix O from the three orthogonal unit eigenvectors. Prove that 0-1 OT. Provide the relation between x and x and confirm that Q(1, 2, 3) Q(x1, x2, x3). = =
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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Transcribed Image Text:A2.
Find the symmetric matrix A associated with the quadratic form
Q(x1, x2, x3) = 2x1x2 + 2x1x3 + 2x2x3.
Use the Principal Minor Test to establish whether Q is positive/negative (semi-)definite, or
indefinite.
Find the eigenvalues -1,-1,2 of the above quadratic forn Q and classify Q based on these
eigenvalues. Are the two classifications, via the PMT and the eigenvalues, consistent? Find
the corresponding unit eigenvectors. There will be two eigenvectors for the repeated eigen
value A1,2 = -1; show that one can choose these such that the columns of O are othogo-
nal. Construct the orthogonal matrix O from the three orthogonal unit eigenvectors. Prove
that 0-¹ OT. Provide the relation between x and x and confirm that Q(x1, x2, ã3)
Q(x1, x2, x3).
=
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