1 Consider the symmetric matrix A = 1 a 2 a 4 a) Using the quadratic form Q(x) = x² Ax determine the value of the element a for which A is not full-rank, i.e. it has at least one eigenvalue equal to 0. b) From the quadratic form of the completed matrix A (i.e. using the value a from the previous part) determine the rank and the dimension of the null space. c) Is the completed matrix positive semi-definite, negative semi-definite or indefinite? d) Compute the null space of the completed matrix A. Can you think of different ways how to obtain the null space of a matrix?
1 Consider the symmetric matrix A = 1 a 2 a 4 a) Using the quadratic form Q(x) = x² Ax determine the value of the element a for which A is not full-rank, i.e. it has at least one eigenvalue equal to 0. b) From the quadratic form of the completed matrix A (i.e. using the value a from the previous part) determine the rank and the dimension of the null space. c) Is the completed matrix positive semi-definite, negative semi-definite or indefinite? d) Compute the null space of the completed matrix A. Can you think of different ways how to obtain the null space of a matrix?
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:1
Consider the symmetric matrix A
=
1
a
2
a
4
a) Using the quadratic form Q(x) = x² Ax determine the value of the element a for which A is
not full-rank, i.e. it has at least one eigenvalue equal to 0.
b) From the quadratic form of the completed matrix A (i.e. using the value a from the previous
part) determine the rank and the dimension of the null space.
c) Is the completed matrix positive semi-definite, negative semi-definite or indefinite?
d) Compute the null space of the completed matrix A. Can you think of different ways how to
obtain the null space of a matrix?
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