# Why won't Mathematica obtain eigenvectors for this symmetric matrix?

The matrix is a real symmetric 4x4:

({{0, Sin[x + y], Sin[z + y], Sin[x + z]},
{Sin[x + y], 0, Sin[x - z], Sin[z - y]},
{Sin[z + y], Sin[x - z], 0, Sin[y - x]},
{Sin[x + z], Sin[z - y], Sin[y - x], 0}})


Mathematica can find the eigenvalues fine, and they are degenerate for x, y, and/or z equal to zero.

No eigenvectors can be found: Eigenvectors::eivec0: Unable to find all eigenvectors. And it seems to me that Mathematica should be able to handle the degeneracy, but I would be interested to know why it is encountering trouble.

Is there some obvious property about this matrix that I am not accounting for (such as restricting values of x, y, z for the trigonometric functions) or is there something I can do inside of Mathematica to specify some starting point for an eigenvalue?

• Setting your matrix equal to mat, Eigenvectors[mat] returns a large output and Eigenvectors[mat,1] returns what is presumably an eigenvector. (This is in v10.) Can you explain what you are doing to generate the error? Jul 21 '14 at 18:02
• ...and v9 does appear to struggle with Eigenvectors[mat,1], let along finding all Eigenvectors, which I did not attempt yet. Jul 21 '14 at 18:04
• I am using v9, there's not much I can do about that. All I did was Eigenvectors[mat], and it returns the error message and four zero eigenvectors. I realize the closed form output will be large, but ultimately I will pick out pieces of the expressions for other things (e.g. the Root pieces proving whether values of x, y, z can produce imaginary eigenvectors). Any idea why v9 struggles? Jul 21 '14 at 19:50

Perhaps your systerm is timing out.

\$Version


"10.0 for Mac OS X x86 (64-bit) (June 29, 2014)"

mat = ({{0, Sin[x + y], Sin[z + y], Sin[x + z]}, {Sin[x + y], 0, Sin[x - z],
Sin[z - y]}, {Sin[z + y], Sin[x - z], 0, Sin[y - x]}, {Sin[x + z],
Sin[z - y], Sin[y - x], 0}});

eval = Eigenvalues[mat] // Simplify;

Length[eval]


4

eval /. {x -> 0, y -> 0, z -> 0}


{0, 0, 0, 0}

Timing[evec = Eigenvectors[mat] // Simplify;]


{107.624447, Null}

Length[evec]


4

The symbolic eigenvectors are very complicated. For example, just the first eigenvector (simplified) is

evec[]


{-((3 Sin[x - 3 y] - 3 Sin[3 x - y] - 5 Sin[x + y] + Sin[3 (x + y)] + Sin[3 x + y - 2 z] + Sin[x + 3 y - 2 z] + 4 Sin[x + y] [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) + 2 Sqrt Cos[ x - y - 2 z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - 2 Sqrt Cos[ x - y + 2 z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + Sin[3 x + y + 2 z] + Sin[x + 3 y + 2 z])/(Sin[2 x + y - 3 z] - 5 Sin[y - z] + Sin[3 (y - z)] + Sin[2 x + 3 y - z] + 4 Sin[y - z] [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) - 2 Sqrt Cos[ 2 x - y - z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt Cos[ 2 x + y + z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - Sin[2 x - 3 y + z] - 3 Sin[3 y + z] - Sin[2 x - y + 3 z] + 3 Sin[y + 3 z])), (8 Cos[x] Cos[y]^2 Sin[x] - 8 Cos[x] Cos[z]^2 Sin[x] - 8 Cos[x]^2 Cos[y] Sin[y] + 8 Cos[y] Cos[z]^2 Sin[y] + 8 Cos[y] Sin[x]^2 Sin[y] - 8 Cos[x] Sin[x] Sin[y]^2 - 8 Cos[x]^2 Cos[z] Sin[z] + 8 Cos[y]^2 Cos[z] Sin[z] + 8 Cos[z] Sin[x]^2 Sin[z] - 8 Cos[z] Sin[y]^2 Sin[z] + 8 Cos[x] Sin[x] Sin[z]^2 - 8 Cos[y] Sin[y] Sin[z]^2 - 2 Sqrt Sin[2 x] Sin[ 2 y] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt Sin[2 x] Sin[ 2 z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt Sin[2 y] Sin[ 2 z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))))/(Sin[ 2 x + y - 3 z] - 5 Sin[y - z] + Sin[3 (y - z)] + Sin[2 x + 3 y - z] + 4 Sin[y - z] [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) - 2 Sqrt Cos[ 2 x - y - z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt Cos[ 2 x + y + z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - Sin[2 x - 3 y + z] - 3 Sin[3 y + z] - Sin[2 x - y + 3 z] + 3 Sin[y + 3 z]), -((Sin[x - 2 y - 3 z] + Sin[x + 2 y - 3 z] - 5 Sin[x - z] + Sin[3 (x - z)] + Sin[3 x - 2 y - z] + Sin[3 x + 2 y - z] + 4 Sin[x - z] [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) + 2 Sqrt Cos[ x - 2 y + z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - 2 Sqrt Cos[ x + 2 y + z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - 3 Sin[3 x + z] + 3 Sin[x + 3 z])/(Sin[2 x + y - 3 z] - 5 Sin[y - z] + Sin[3 (y - z)] + Sin[2 x + 3 y - z] + 4 Sin[y - z] [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2)) - 2 Sqrt Cos[ 2 x - y - z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) + 2 Sqrt Cos[ 2 x + y + z] [Sqrt](3 - Cos[2 x] Cos[2 y] - Cos[2 x] Cos[2 z] - Cos[2 y] Cos[ 2 z] - [Sqrt](3 - 2 Cos[4 x] - 2 Cos[4 y] - 2 Cos[2 y] Cos[2 z] + Cos[2 y]^2 Cos[2 z]^2 + Cos[2 x]^2 (Cos[2 y] + Cos[2 z])^2 - 2 Cos[4 z] - 2 Cos[2 x] (Cos[2 z] Sin[2 y]^2 + Cos[2 y] Sin[2 z]^2))) - Sin[2 x - 3 y + z] - 3 Sin[3 y + z] - Sin[2 x - y + 3 z] + 3 Sin[y + 3 z])), 1}