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In Mathematica, I'm trying to find the eigenvalues and eigenvectors of a 10x10 matrices that is diagonalizable for sure.

The matrix ix:

{{0, 0, 0, 0, -2 t, -2 t, -2 t, -2 t, 0, 0}, {0, 0, 0, 0, -t, t, -t,

t, 0, 0}, {0, 0, 2 U, 0, -t, t, -t, t, 0, 0}, {0, 0, 0, 2 U, -2 t,

2 t, -2 t, 2 t, 0, 0}, {-Sqrt[2] t, -t, -t, -Sqrt[2] t, U, 0, 0,

0, -t, -t}, {-Sqrt[2] t, t, t, Sqrt[2] t, 0, U, 0,

0, -t, -t}, {-Sqrt[2] t, -t, -t, -Sqrt[2] t, 0, 0, U,

0, -t, -t}, {-Sqrt[2] t, t, t, Sqrt[2] t, 0, 0, 0, U, -t, -t}, {0,

0, 0, 0, -t, -t, -t, -t, U, 0}, {0, 0, 0, 0, -t, -t, -t, -t, 0, U}}

where t and U are both Reals.

The command I use is: Assuming[t \[Element] Reals,

Assuming[U \[Element] Reals, Eigensystem[MATRIXNAME]]]

I'm getting some symbols that I don't know. For example, one of the eigenvalues is:

Root[8 t^2 U - 8 t^2 #1 - 8 Sqrt[2] t^2 #1 + 2 U^2 #1 -

3 U #1^2 + #1^3 &, 1]

What are these # and &? What does mean "Root[expression, 1]" ?

Regards,

Marcus.