I have a Hermitian 4x4 matrix $H$ which has 2 positive and 2 negative eigenvalues with orthogonal eigenvectors. Let $P$ be the projector to the 2-plane given by the two eigenvectors with positive eigenvalues. I wish to find the restriction to positive eigenvalues $H'$, which I define as $$ H'=PHP=HP=PH $$ The matrix $H'$ is then equal to $H$ on positive eigenvectors and to $0$ on the negative eigenvectors.
I wish to study the behaviour of $H'$ (or $P$) as $H$ varies. The definition that I gave above is fine when the two positive eigenvectors are non-degenerate. But if I am studying $H'$ near $H_0$, where the positive eigenvectors of $H_0$ are degenerate, there is a singularity of the eigenvectors, i.e. the eigenvectors components are not continuous with respect to continuous variations of $H$ around $H_0$. But $P$ and $H'$ themselves are not singular around this point, the singularity is only an artefact of the way that I am computing it (via first computing the singular eigenvectors, which I forget about later).
Is there a better way to define or compute the $H'$ or $P$, without resorting to the eigenvectors?
Example: In 2 dimensions, if $$ H=a_0+\vec{a}\cdot \vec{\sigma} \,, $$ where $\vec{\sigma}$ is the 3-vector of the Pauli matrices, and $a_0$ is a number implicitly multiplied by the unit matrix. The $a_0$ and $\vec{a}$ are linear in the components of $H$ and can be computed easily by traces: $a_0=\text{Tr} H$ and $\vec{a}=\text{Tr} H\vec{\sigma}$. Then I can write $$ P=1+\frac{\vec{a}}{|\vec{a}|}\cdot \vec{\sigma} \,, \qquad H'=\frac{a_0+|\vec{a}|}{2} \left(1+\frac{\vec{a}}{|\vec{a}|}\cdot \vec{\sigma}\right) $$ which is continuous for the cases for which we define $P$ and $H'$, i.e. for $|\vec{a}|>|a_0|$, so the matrix is semi-definite. Of course in 2 dimensions, there is no degeneracy and the eigenvectors are also continuous, but this is an example of a solution which would be satisfactory, if it could be generalized to 4x4 matrices.
A satisfactory solution would also be computing the $P$ or $H'$ to first order in the variations of the elements of $H$ around the degenerate $H_0$.
I am actually interested in the case where $H$ is not necessarily Hermitian, the eigenvalues are 2 with positive real part and 2 with negative real part, and the eigenvectors need not be perpendicular. But this is an easier problem which I also can't solve, so I think it is better to start here.
Maybe this problem has already been solved, but I have no idea where to look. I will be grateful for all answers or pointing me in the right direction.