# Are there explicit formulas for the eigenvalues and eigenvectors of a generic 4x4 density matrix?

I have a 4x4 density matrix all of whose elements are nonzero. Its form is $$\begin{pmatrix} a & b & c & d \\ b^* & e & f & g \\ c^* & f^* & h & j \\ d^* & g^* & j^* & k \end{pmatrix}$$ where $a+e+h+k=1$, or in block form

$$\begin{pmatrix} A & B \\ B^\dagger & C \end{pmatrix}.$$

Is there simple way to find the eigenvalues and eigenvectors of this matrix?

I don't want to the largest or smallest eigenvalues. All the elements of the matrix are not a number. I calculated by help of the Mathematica and Maple (too much terms) but I want an analytical method.

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• How is this a physics question? (It appears to be pure linear algebra that gains nothing from the physics context) What about the determination of eigenvalues e.g. as roots of the characteristic polynomial doesn't satisfy you in this case? – ACuriousMind May 30 '16 at 13:29
• The four solutions $\lambda_{1,2,3,4}$ are real numbers because the matrix is Hermitian. For allowed density matrices, all four roots are non-negative, too. They are the solutions of the equation $\det(\rho - \lambda\cdot {\bf 1})=0$. It is a fourth order equation in $\lambda$ and a very complicated analytic expression exists to solve 4th order equation (for 5th order, it already doesn't exist). Mathematica and Maple returned many terms because the solution for a generic enough 4th order equation has many terms, indeed. It's much worse than e.g. a quadratic equation. – Luboš Motl May 30 '16 at 13:35
• I derived the analytic formula for the quartic equation when I was 17 but it was many, many pages. ;-) The secret is to know how to break the symmetry between the roots and finally identify them separately. – Luboš Motl May 30 '16 at 13:36
• Here's the explicit formula for the solutions of a quartic equation that Luboš alludes to: en.wikipedia.org/wiki/… – Rahul May 31 '16 at 0:12