For each of these matrices:
Find its leading principal minors.
Determine whether it’s positive definite, positive semi-definite, both, or neither.
Find the spectral decomposition. [Hint: An eigenvalue of is 1, and of is 9.]
Find a matrix square root.
Let be a real symmetric matrix, and let be eigenvectors for eigenvalues of , respectively. Prove that , using the standard inner product. Hence deduce that if then and are orthogonal to each other.
Find a symmetric matrix in which has no eigenvalues in .
Write down a non-zero matrix of a symmetric bilinear form, whose entries are all non-negative, but which is not positive semi-definite.
Prove that every real orthogonal matrix is either a rotation or a reflection. In other words, prove that if and then either
or
Let be a skew-symmetric matrix over a field ; assume that is invertible. Define
Prove that is an orthogonal matrix; i.e. prove that .
A student is asked to prove that a real symmetric matrix has only real eigenvalues. His proof goes as follows:
[Student box]
By the fundamental theorem of algebra, the characteristic polynomial has complex roots; in other words, there is a complex number such that . Then we can choose an eigenvector with eigenvalue . Then , and since is real, when we conjugate both sides: . Therefore
Therefore , so .
[End of Student box]
This solution would not get full marks. What are the problems with this solution, and how could they be fixed?
Learning objectives for Chapter 5:
Pass Level: You should be able to…
Determine whether or not a given matrix is orthogonal (e.g. Exercise 5.2).
Correctly answer, with full justification, at least 50% of the true / false questions relevant to this Chapter.
First class level: You should be able to…
Write a complete solution, without referring to any notes, to at least 80% of the exercises in this Chapter, and in particular the proof questions.
Correctly answer, with full justification, all of the true / false questions relevant to this Chapter.