Krylov Space dimension for specific matrix

Question

This was an exam question and I couldn't solve it so I'd like to know what the solution is.

Let $b,c\in \mathbb{R}^n$ be linearly independent and define $$A(b,c)=I+bb^T+cc^T.$$

1. Show that $A(b,c)$ is positive definite
2. Show the Krylov space $\mathcal{K}_l(b,A)$ has dimension atmost $2$ for all natural $l$.
3. For what $b,c$ does $\mathcal{K}_2(b,A)$ have dimension $2$.
4. For $x^0$ what is the minimum and maximum number of iterations for the CG method for $A(b,c) x=b$.

Solution attempt:

1. We have for $y\in \mathbb{R}^n\setminus\{ 0\}$: $$y^T A(b,c) y= y^Ty+ y^Tbb^Ty+y^T cc^Ty=y^Ty+(y^Tb)^2+(y^Tc)^2 \geq y^Ty > 0.$$ and the above is $0$ only when $y^T$ is.
2. $\mathcal{K}_1=\text{span}( b)$ and $\mathcal{K}_2=\text{span}(b,A(b,c)b)$. Set $bb^T=B$ and $cc^T=C$. $$Ab=(I+B+C)b=b+Bb+Cb=b+bb^Tb+cc^Tb=b+\|b\|^2b+\langle c,b\rangle c.$$

$$A^2b=(I+B+C)(b+Bb+Cb)=b+2Bb+2Cb+B^2b+C^2b+BCb+CBb$$ I have to show $\{b,Ab,A^2b\}$ is linearly dependent but I'm not getting anywhere.

3. We want $b$ and $b+Bb+Cb$ to be linearly independent.
4. Maximum number of iterations in general is $n$ but here it seems to be different.

Answer 1

You need to notice that \begin{align} Ab&=b+(b^Tb)b+(c^Tb)c\\ Ac&=c+(b^Tc)b+(c^Tc)c\\ \end{align} so that all Krylov spaces are spanned by the vectors $b$ and $c$. Now consider what happens when these vectors are orthogonal.