How can I Reduce the covariance Matrix?

Question

How can I reduce this covariance matrix? And also plot its eigenvalues? and how to compute its eigenvectors? Can we compute its eigenvalues and eigenvectors without reducing in compact form?

{{x + Cosh[2*t], -I, (x + Sinh[2*t])/Sqrt[2], 0, -((-3*x + Sinh[2*t])/Sqrt[2]), 0}
,{I, x + Cosh[2*t], 0, (x + Sinh[2*t])/Sqrt[2], 0, (-3*x + Sinh[2*t])/Sqrt[2]}
,{(x + Sinh[2*t])/Sqrt[2], 0, (1 + x + Cosh[2*t])/2, -I, (1 + 3*x - Cosh[2*t])/2, 0}
,{0, (x + Sinh[2*t])/Sqrt[2], I, (1 + x + Cosh[2*t])/2, 0, (-1 - 3*x + Cosh[2*t])/2}
,{-((-3*x + Sinh[2*t])/Sqrt[2]), 0, (1 + 3*x - Cosh[2*t])/2, 0, (1 + 9*x + Cosh[2*t])/2,-I}
,{0, (-3*x + Sinh[2*t])/Sqrt[2], 0, (-1 - 3*x + Cosh[2*t])/2, I, (1 + 9*x + Cosh[2*t])/2}
}

Answer 1

Here's a shot at plotting the eigenvalues:

evs = Eigenvalues[m]

{1/2 Root[-64 x-80 x Cosh[2 t]-48 x Sinh[2 t]+(4+44 x+12 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-4-12 x-4 Cosh[2 t]) #1^2+#1^3&,1]

,1/2 Root[-64 x-80 x Cosh[2 t]-48 x Sinh[2 t]+(4+44 x+12 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-4-12 x-4 Cosh[2 t]) #1^2+#1^3&,2]

,1/2 Root[-64 x-80 x Cosh[2 t]-48 x Sinh[2 t]+(4+44 x+12 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-4-12 x-4 Cosh[2 t]) #1^2+#1^3&,3]

,1/2 Root[-16-48 x+16 Cosh[2 t]+48 x Cosh[2 t]+16 x Sinh[2 t]+(-4-28 x+4 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-12 x-4 Cosh[2 t]) #1^2+#1^3&,1]

,1/2 Root[-16-48 x+16 Cosh[2 t]+48 x Cosh[2 t]+16 x Sinh[2 t]+(-4-28 x+4 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-12 x-4 Cosh[2 t]) #1^2+#1^3&,2]

,1/2 Root[-16-48 x+16 Cosh[2 t]+48 x Cosh[2 t]+16 x Sinh[2 t]+(-4-28 x+4 Cosh[2 t]+40 x Cosh[2 t]+8 x Sinh[2 t]) #1+(-12 x-4 Cosh[2 t]) #1^2+#1^3&,3]}

Plot3D[Evaluate@evs, {x, -2, 2}, {t, 0, 2},
  PlotRange -> {-10, 20},  AxesLabel -> {"x", "t"},
  PlotStyle -> Opacity[0.3], Mesh -> None
]