Dear Nichlas Vous Christensen,

From the first part of Caylay-Hamilton theorem, for a 3x3 matrix A, it is easy to find characteristic polynom under the following equation :

LaTeX on : $-\lambda^3 + tr(A) \cdot \lambda^2 - \left ( \sum_{i=1}^3 det(M_{i ; i}(A)) \right ) \cdot \lambda + det(A)$

tr(A) = trace of A, M_{i ; i}(A) minor of A along column i and row i, det(A) its determinant (to add : \left ( \sum_{i=1}^3 det(M_{i ; i}(A)) \right ) = tr(A^{-1})).

Then, it can be very harsh to find solutions for a third degree polynom... Have you tried using Cardan method to solve your problem ? I think, unfortunately, it will be a long process on a piece of sheet...

Dear Nichlas Vous Christensen,

It is quite easy to find the expression of characteristic polynom of a 3x3 matrix A using the first step of Caylay-Hamilton theorem :

LaTeX on : P_A = -\lambda^3 + \textrm{tr}(A) \lambda^2 - \underbrace{\left ( \sum_{i=1}^3 \det (M_{i ; i}(A)) \right )}_{\textrm{tr}(A^{-1})} \lambda + \det(A).

With det(A), determinant of A, \lambda an eigenvalue, M_{i ; i} the minor of A along column i and row i and tr(A) the trace of A.

Resolution of third degree equation can be very hard, especially in your case on a piece of sheet... But not impossible. Did you try to use Cardan method to solve your equation ?

a 3x3 will give a cubic equation which is possible to solve directly (though tricky).

The standard method for anything greater than a 2x2 is to diagonalize (iteratively) the matrix. You may find my paper in J magn reson 1986 useful or even my paper

" Two-dimensional Two Contact Double Resonance Spectroscopy" in Z. Natfor. 55a useful.

Dear Nichlas,

you may want to take a look at this paper:

Article Longitudinal relaxation in dipole-coupled homonuclear three-...

To my understanding the authors tell, that due to cross-correlated relaxation your description of the relaxation of a 3-spin system (involving only Iz magnetisation) might not be adequate to describe the situation in full detail....

Maybe the paper and my comment help...

Kind regards

Alfred

P.S. I am aware that this does not answer your question and the comments of the other colleagues above might be more adequate!

Some related papers:

• DOI: 10.1016/S0009-2614(00)00182-2
• DOI: 10.1021/ja00069a077
• DOI: 10.1063/1.460866
• Proceedings of the Indian Academy of Sciences - Section A volume 54, pages13–23(1961)