*Suggestion:* Try working with a concrete example of a 3×3 matrix with these eigenvalues. See what kind of behavior is has in the context of your question, then see whether you can generalize your result to an arbitrary 3×3 matrix with these eigenvalues.

So, in particular, the 3×3 matrix *A* given by

A :=

[ 7.4 0 0 ]

[ 0 0.7 0 ]

[ 0 0 2.3 ]

has the indicated eigenvalues, with corresponding eigenvectors **v**\_1 := [1 0 0]^(T), **v**\_2 := [0 1 0]^(T), and **v**\_3 := [0 0 1]^(T). (I.e., the eigenvectors are the column vectors that are the transposes of the row vectors [1 0 0], [0 1 0], and [0 0 1], respectively.)

A natural starting place would be to consider, for each *j*, the limits

- lim\_[*t*→∞] *A*^(*t*) **v**\_*j*

over the eigenbasis vectors {**v**\_1, **v**\_2, **v**\_3}.

After working with this example, of course, you'd still need to consider the general case. Can you show how this specific case is enough to describe the behavior of the general case?

I hope something here points you in a useful direction. Good luck!