While all frames of references are as valid as each other, it is very hard to place one's self in a photon's frame of reference.
Indeed, a photon has no rest frame. It travels at a speed c, which is a singularity for the Lorentz equations which allow us to relate the frame of one ponderable bit of matter with another.
A 3+1 Lorentz transform can be parameterized either by a dimensionless ratio with the speed of light, $$\beta = \frac{v}{c}$$, or by a dimensionless rapidity parameter, $$\rho = \tanh^{-1} \beta$$.
$${\Huge \Lambda}(\rho, \hat{x}) = \begin{pmatrix} c^{-1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} e^{-\rho} & 0 & 0 & 0 \\ 0 & e^{\rho} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} c & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\\ \quad \quad \quad = \begin{pmatrix} \cosh \rho & c^{-1} \sinh \rho & 0 & 0 \\ c \sinh \rho & \cosh \rho & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 - \beta^2}} & \frac{\beta}{c \sqrt{1 - \beta^2}} & 0 & 0 \\ \frac{c \beta}{\sqrt{1 - \beta^2}} &\frac{1}{\sqrt{1 - \beta^2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = {\Huge \Lambda}(\beta, \hat{x})$$
$$e^{\rho} = \frac{\sqrt{1 + \beta}}{\sqrt{1 - \beta}}$$
For $$|\beta| < 1$$ we have $$\Lambda(-\beta) \Lambda(\beta) = 1$$, $$\det \Lambda = 1$$ and $$\left( \Lambda u\right)^T \eta \Lambda v = u^T \eta v$$ for any two space-time vectors, u and v, which are all physically important.*
But $$\lim_{|\beta| \to 1} \cosh \rho = \lim_{|\beta| \to 1} \frac{1}{\sqrt{1 - \beta^2}} = \infty$$. This represents a singularity in the transform to-and-from any purported frame of the photon.
As for the eigenvectors, one goes to zero while another goes to infinity, making the situation worse than a typical singular matrix.
* $$\eta \propto \begin{pmatrix} -c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$, this is the metric of flat space-time in Cartesian coordinates.
//edited to fix math display
That's my favourite word!