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Exercises

Exercise 5.25:

$A:=\begin{bmatrix}3&1\\ 1&3\end{bmatrix}$
$B:=\begin{bmatrix}4&-2\\ -2&7\end{bmatrix}$
$C:=\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&2\end{bmatrix}$
$D:=\begin{bmatrix}1&0&0\\ 0&1&-3\\ 0&-3&9\end{bmatrix}$
$E:=\begin{bmatrix}1&3&1\\ 3&10&0\\ 1&0&10\end{bmatrix}$
$F:=\begin{bmatrix}2&-1&-1\\ -1&2&1\\ -1&1&4\end{bmatrix}$
$G:=\begin{bmatrix}2&-2&6\\ -2&-3&-4\\ 6&-4&1\end{bmatrix}$

For each of these matrices:

i.

Find its leading principal minors.
ii.

Determine whether it’s positive definite, positive semi-definite, both, or neither.
iii.

Find the spectral decomposition. [Hint: An eigenvalue of $F$ is 1, and of $G$ is 9.]
iv.

Find a matrix square root.

Exercise 5.26:

Let $A$ be a real symmetric matrix, and let $\vec{x},\vec{y}$ be eigenvectors for eigenvalues $\lambda,\mu$ of $A$ , respectively. Prove that $\lambda\langle\vec{x},\vec{y}\rangle=\mu\langle\vec{x},\vec{y}\rangle$ , using the standard inner product. Hence deduce that if $\lambda\neq\mu$ then $\vec{x}$ and $\vec{y}$ are orthogonal to each other.

Exercise 5.27:

Find a symmetric matrix in $\operatorname{M}_{{2}}({{\mathbb{Q}}})$ which has no eigenvalues in ${\mathbb{Q}}$ .

Exercise 5.28:

Write down a non-zero matrix of a symmetric bilinear form, whose entries are all non-negative, but which is not positive semi-definite.

Exercise 5.29:

Prove that every $2\times 2$ real orthogonal matrix is either a rotation or a reflection. In other words, prove that if $A\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ and $A^{T}A=I_{2}$ then either

A=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}

A=\begin{bmatrix}\cos\theta&\sin\theta\\ \sin\theta&-\cos\theta\end{bmatrix}.

Exercise 5.30 (Cayley’s formula):

Let $S\in\operatorname{M}_{{n}}({F})$ be a skew-symmetric matrix over a field $F$ ; assume that $S-I_{n}$ is invertible. Define

P:=(S-I_{n})^{-1}(S+I_{n}).

Prove that $P$ is an orthogonal matrix; i.e. prove that $P^{T}P=I_{n}$ .

Exercise 5.31:

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 $c_{A}$ has complex roots; in other words, there is a complex number $\lambda\in{\mathbb{C}}$ such that $c_{A}(\lambda)=0$ . Then we can choose an eigenvector $x\in{\mathbb{C}}^{n}$ with eigenvalue $\lambda$ . Then $Ax=\lambda x$ , and since $A$ is real, when we conjugate both sides: $A\overline{x}=\overline{\lambda}\overline{x}$ . Therefore

\lambda x^{T}\overline{x}=(Ax)^{T}\overline{x}=x^{T}A\overline{x}=x^{T}(% \overline{\lambda}\overline{x})=\overline{\lambda}x^{T}\overline{x}.

Therefore $\lambda=\overline{\lambda}$ , so $\lambda\in{\mathbb{R}}$ .

[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).
•

Test whether or not a real symmetric matrix is positive definite, or positive semi-definite (e.g. Exercise 5.25(ii)).
•

Compute the spectral decomposition $PDP^{T}$ of a real symmetric matrix (e.g. Exercise 5.25(iii)).
•

Use a spectral decomposition to find a matrix square root of a real symmetric matrix (e.g. Exercise 5.25(iv)).
•

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.