# How can I find eigen values and eigen vectors of a symbolic matrix? [closed]

How can I find eigenvalues and eigenvectors of this system given as a matrix?

 {{x + Cosh[2 t], -I, (x + Sinh[2 t])/Sqrt[2], 0, (x + Sinh[2 t])/Sqrt[2], 0, 0, 0},
{I, x + Cosh[2 t], 0, (x + Sinh[2 t])/Sqrt[2], 0, -((x + Sinh[2 t])/Sqrt[2]), 0, 0},
{(x + Sinh[2 t])/Sqrt[2], 0, 1/2 (1 + x + Cosh[2 t]), -I, 1/2 (1 + x + Cosh[2 t]), 0, 0, 0},
{0, (x + Sinh[2 t])/Sqrt[2], I, 1/2 (1 + x + Cosh[2 t]), 0, 1/2 (-1 - x - Cosh[2 t]), 0, 0},
{(x + Sinh[2 t])/Sqrt[2], 0, 1/2 (1 + x + Cosh[2 t]), 0, 1/2 (1 + x + Cosh[2 t]), -I, 0, 0},
{0, -((x + Sinh[2 t])/Sqrt[2]), 0, 1/2 (-1 - x - Cosh[2 t]), I, 1/2 (1 + x + Cosh[2 t]), 0, 0},
{0, 0, 0, 0, 0, 0, 1, I}, {0, 0, 0, 0, 0, 0, -I, 1}}

• Have you tried Eigensystem? Commented Jul 17, 2018 at 7:46
• yes but a very large output is taken out Commented Jul 17, 2018 at 7:48
• @Henrik Schumacher sir can you help me regarding this that what can I do for that system I want the answer in the compact form Commented Jul 17, 2018 at 7:50
• Yeah, that's what is about to happen with symbolic computations. Note that finding the eigenvalues amounts to finding the roots of the charakteristic polynomial. Since your matrix has size $8 \times 8$, this polynomial is of degree $8$, hence a closed form expression for all its roots need not exist. That's why certain Root expressions show up. Commented Jul 17, 2018 at 7:50
• Probably, nobody can help you here: Your problem is not that the result wasn't correct; your problem is that you don't like the result. Commented Jul 17, 2018 at 7:53

If you are looking only for numerical solution, just defer the evaluation of Eigensystem until numerical data is present. You can use this function for that:

eigsys[x_, t_] := Eigensystem[N[
{{x + Cosh[2 t], -I, (x + Sinh[2 t])/Sqrt[2],
0, (x + Sinh[2 t])/Sqrt[2], 0, 0, 0}, {I, x + Cosh[2 t],
0, (x + Sinh[2 t])/Sqrt[2], 0, -((x + Sinh[2 t])/Sqrt[2]), 0,
0}, {(x + Sinh[2 t])/Sqrt[2], 0, 1/2 (1 + x + Cosh[2 t]), -I,
1/2 (1 + x + Cosh[2 t]), 0, 0, 0}, {0, (x + Sinh[2 t])/Sqrt[2],
I, 1/2 (1 + x + Cosh[2 t]), 0, 1/2 (-1 - x - Cosh[2 t]), 0,
0}, {(x + Sinh[2 t])/Sqrt[2], 0, 1/2 (1 + x + Cosh[2 t]), 0,
1/2 (1 + x + Cosh[2 t]), -I, 0,
0}, {0, -((x + Sinh[2 t])/Sqrt[2]), 0, 1/2 (-1 - x - Cosh[2 t]),
I, 1/2 (1 + x + Cosh[2 t]), 0, 0}, {0, 0, 0, 0, 0, 0, 1,
I}, {0, 0, 0, 0, 0, 0, -I, 1}}
]];

• sir this is not working in Mathematica Commented Jul 17, 2018 at 8:42
• What do you mean, it doesn't work? Which version of Mathematica do you use? Commented Jul 17, 2018 at 10:00