This is from Hartley-Zisserman's Multiple View Geometry in Computer Vision pg 55-56.
We are given a conic $$C_\infty^* = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 0 \end{bmatrix}$$ and we know that there is some unknown projectivity $H \in PGL(3, \mathbb{R})$ such that $C_\infty^{'*} = HC_\infty^*H^T$. The book goes on to claim that we can write the SVD of $C_\infty^{'*}$ as $$C_\infty^{'*} = U \begin{bmatrix} 1 & 0 & 0 \\ 0& 1 & 0 \\ 0& 0 & 0 \end{bmatrix} U^T$$ and then $U = H$ up to a similarity transformation.
This is the part I don't understand. The decomposition above is just a orthogonal diagonalization of a symmetric matrix $C_\infty^{'*}$ but there reason apriori that $C_\infty^{'*}$ has the same eigenvalues $1,1,0$ as $C_\infty^{'*}$ is obtained from $C_\infty^*$ by a congruence rather than a similarity relation. For example if $H = \verb|diag|(1,2,3)$, we have $HC_\infty H^T= \verb|diag|(1,4,0)$. Can you help me see why this is true, and if it is not true, help me see what the author meant here?
Also a side comment is that the Hartley/Zisserman book states many results without proof or comment or citation, and I find it quite difficult each time I stumble on something. I would say that I have a reasonable background in Linear algebra, so I don't think the issue is that I'm not prepared for this book. Any advice on how to tackle this issue would be appreciated. Is there another book which has all these details?
Take $C^*_\infty$ and ${C^*_\infty}^\prime$ the dual conics of the points $x$ and $x^\prime$ related by the projection $H$. The two conics have the same eigenvalues because $C^*_\infty$ is a degenerate conic therefore it has rank 2 and has a repeated point (pg 32). Moreover, $l_\infty$ is in the null space of $C^*_\infty$ and we have
$$ {C^*_\infty}^\prime {l_\infty}^\prime = HC^*_\infty H^t l_\infty = HC_\infty^* l_\infty = 0 $$
Therefore $l_\infty$ is also in the null space of ${C_\infty^*}^\prime$. If $p$ is the repeated point of $C^*_\infty$ (with eigenvalue 1) we have $C^*_\infty p = p$ and $p^\prime = H^{-T}p = H p$ and we can write
$$ {C^*_\infty}^\prime {p}^\prime = HC^*_\infty H^t H^{-T}p = HC_\infty^* p = Hp = p^\prime $$
So $p$ is also a repeated point of $C^*_\infty$.
About your question about a book having all these details, I think you cant take a look at two books by Olivier Fagueras that are related to the subject of Hartley and Zisserman book. These are: