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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?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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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?
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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