# How to get eigenvectors of a 4x4 matrix? [closed]

 MatrixForm[m = {{2, 9, 0}, {3, 8, 9}, {3, 9, 1}}]
Eigensystem[m]


I am facing problem in finding eigenvectors using Eigensystem. I have tried the already given answers in the forum but failed please help

The output is full of Root functions.

• Could you be more specific what's causing problems? It will be hard for people to say anything without a concrete example. Commented Jun 12, 2019 at 13:24
• Title says 4x4, in the question is a 3x3 example. So what are you really after? And the example works totally fine: imgur.com/a/fAq6in3 Commented Jun 12, 2019 at 14:20
• See (13767); also related is (42234). Commented Jun 12, 2019 at 14:30
• Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! Commented Jun 12, 2019 at 14:30
• You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find this meta Q&A helpful Commented Jun 12, 2019 at 14:31

Try the option Eigensystem[m, Quartics -> True] (okay, the docs say to use the Cubics one, too, but I haven't found it necessary yet):

SeedRandom[0];
m = RandomInteger[{-2, 2}, {4, 4}];

FreeQ[Root]@Eigensystem[m]
(*  False  *)

FreeQ[Root]@Eigensystem[m, Quartics -> True]
(*  True  *)


For bigger than 4x4, you run into problems expressing the roots of the characteristic polynomial in terms of radicals, as pointed out in this comment to Eigensystem without Root objects

Let's generate a random 4x4 matrix:

a = RandomReal[{-10, 10}, {4, 4}];
MatrixForm[a]


Let's compute the eigenvalues and eigenvectors with Eigensystem. The result is a list that contains both:

{spec, vect} = Eigensystem[a]


Let's check that the first eigenvector is indeed associated to the first eigenvalue

a.vect[[1]] == spec[[1]] vect[[1]]


I will let you do the same for the remaining 3 eigenvectors.

It's all well documented in the help for the Eigensystem function. If you are new to Mathematica, please take a look at the documentation (especially the documentation of each new function you encounter). It is very well written and quite useful on a daily basis, for the beginner as well as for the advanced user.

You should not think of this forum as a substitute for the Mathematica documentation because the topics here are usually more advanced and won't cover the basis you might be looking for :-)