Suppose the following matrix:
$$ M=\pmatrix{a&b&c&d\\e&f&g&h\\i&j&k&k\\m&n&o&p} $$
I am trying to solve this but the problem does not seem to have a short and sweet solution. I was wondering what can be done to improve the elegance of the solution further. Are there any known symmetries that can simplify the solution, for a problem of this type?
For reference, requesting the eigenvalues in Mathematica via
M=({
{a, b, c, d},
{e, f, g, h},
{i, j, k, l},
{m, n, o, p}
});
M//MatrixForm
Part[Eigenvalues[M],1]
ToRadicals[%]
The line M= Part[Eigenvalues[M],1] returns:
Root[d g j m - c h j m - d f k m + b h k m + c f l m - b g l m -
d g i n + c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p + (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n - d i o -
h j o + a l o + f l o + b e p - a f p + c i p + g j p - a k p -
f k p) #1 + (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) #1^2 + (-a - f - k - p) #1^3 + #1^4 &, 1]
And the line ToRadicals[%] returns:
1/4 (a + f + k + p) -
1/2 \[Sqrt](b e - a f + c i + g j - a k - f k + d m + h n + l o +
1/4 (-a - f - k - p)^2 - a p - f p - k p +
1/3 (-b e + a f - c i - g j + a k + f k - d m - h n - l o + a p +
f p + k p) + (2^(
1/3) ((-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j + b e k -
a f k + d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p + a f k p)))/(3 (2 (-b e +
a f - c i - g j + a k + f k - d m - h n - l o + a p +
f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p) (c f i - b g i -
c e j + a g j + b e k - a f k + d f m - b h m + d k m -
c l m - d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p - a k p -
f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n -
a h k n - c e l n + a g l n + d f i o - b h i o -
d e j o + a h j o + b e l o - a f l o - c f i p +
b g i p + c e j p - a g j p - b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p +
a f k p))^3 + (2 (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p) (c f i -
b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o +
b e p - a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p))^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k - d m -
h n - l o + a p + f p + k p) (c f i - b g i - c e j +
a g j + b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o + a l o +
f l o + b e p - a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o + a h j o +
b e l o - a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o + a h j o +
b e l o - a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p + a f k p))^3 + (2 (-b e +
a f - c i - g j + a k + f k - d m - h n - l o + a p +
f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p) (c f i - b g i -
c e j + a g j + b e k - a f k + d f m - b h m + d k m -
c l m - d e n + a h n + h k n - g l n - d i o -
h j o + a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p))^2))^(1/3))) -
1/2 \[Sqrt](b e - a f + c i + g j - a k - f k + d m + h n + l o +
1/2 (-a - f - k - p)^2 - a p - f p - k p +
1/3 (b e - a f + c i + g j - a k - f k + d m + h n + l o - a p -
f p - k p) - (2^(
1/3) ((-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j + b e k -
a f k + d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p + a f k p)))/(3 (2 (-b e +
a f - c i - g j + a k + f k - d m - h n - l o + a p +
f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p) (c f i - b g i -
c e j + a g j + b e k - a f k + d f m - b h m + d k m -
c l m - d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p - a k p -
f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n -
a h k n - c e l n + a g l n + d f i o - b h i o -
d e j o + a h j o + b e l o - a f l o - c f i p +
b g i p + c e j p - a g j p - b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p +
a f k p))^3 + (2 (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p) (c f i -
b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o +
b e p - a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p))^2))^(1/3)) - (1/(
3 2^(1/3)))((2 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k - d m -
h n - l o + a p + f p + k p) (c f i - b g i - c e j +
a g j + b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o + a l o +
f l o + b e p - a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o + a h j o +
b e l o - a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o + a h j o +
b e l o - a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p + a f k p))^3 + (2 (-b e +
a f - c i - g j + a k + f k - d m - h n - l o + a p +
f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p) (c f i - b g i -
c e j + a g j + b e k - a f k + d f m - b h m + d k m -
c l m - d e n + a h n + h k n - g l n - d i o -
h j o + a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p))^2))^(
1/3)) - (-(-a - f - k - p)^3 +
4 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k - d m -
h n - l o + a p + f p + k p) -
8 (c f i - b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p))/(4 \[Sqrt](b e - a f + c i + g j -
a k - f k + d m + h n + l o + 1/4 (-a - f - k - p)^2 -
a p - f p - k p +
1/3 (-b e + a f - c i - g j + a k + f k - d m - h n - l o +
a p + f p + k p) + (2^(
1/3) ((-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m + c f l m -
b g l m - d g i n + c h i n + d e k n - a h k n -
c e l n + a g l n + d f i o - b h i o - d e j o +
a h j o + b e l o - a f l o - c f i p + b g i p +
c e j p - a g j p - b e k p +
a f k p)))/(3 (2 (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p) (c f i -
b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o +
b e p - a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j +
a k + f k - d m - h n - l o + a p + f p +
k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j +
a g j + b e k - a f k + d f m - b h m + d k m -
c l m - d e n + a h n + h k n - g l n - d i o -
h j o + a l o + f l o + b e p - a f p + c i p +
g j p - a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n -
a h k n - c e l n + a g l n + d f i o - b h i o -
d e j o + a h j o + b e l o - a f l o - c f i p +
b g i p + c e j p - a g j p - b e k p +
a f k p))^3 + (2 (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p) (c f i -
b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k -
a f k + d f m - b h m + d k m - c l m - d e n +
a h n + h k n - g l n - d i o - h j o + a l o +
f l o + b e p - a f p + c i p + g j p - a k p -
f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m -
h n - l o + a p + f p + k p) (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p))^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (-b e + a f - c i - g j + a k + f k - d m -
h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k + f k -
d m - h n - l o + a p + f p + k p) (c f i - b g i -
c e j + a g j + b e k - a f k + d f m - b h m +
d k m - c l m - d e n + a h n + h k n - g l n -
d i o - h j o + a l o + f l o + b e p - a f p +
c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o +
b e p - a f p + c i p + g j p - a k p - f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) + \[Sqrt](-4 ((-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^2 -
3 (-a - f - k - p) (c f i - b g i - c e j + a g j +
b e k - a f k + d f m - b h m + d k m - c l m -
d e n + a h n + h k n - g l n - d i o - h j o +
a l o + f l o + b e p - a f p + c i p + g j p -
a k p - f k p) +
12 (d g j m - c h j m - d f k m + b h k m +
c f l m - b g l m - d g i n + c h i n + d e k n -
a h k n - c e l n + a g l n + d f i o - b h i o -
d e j o + a h j o + b e l o - a f l o - c f i p +
b g i p + c e j p - a g j p - b e k p +
a f k p))^3 + (2 (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p)^3 -
9 (-a - f - k - p) (-b e + a f - c i - g j + a k +
f k - d m - h n - l o + a p + f p + k p) (c f i -
b g i - c e j + a g j + b e k - a f k + d f m -
b h m + d k m - c l m - d e n + a h n + h k n -
g l n - d i o - h j o + a l o + f l o + b e p -
a f p + c i p + g j p - a k p - f k p) +
27 (c f i - b g i - c e j + a g j + b e k - a f k +
d f m - b h m + d k m - c l m - d e n + a h n +
h k n - g l n - d i o - h j o + a l o + f l o +
b e p - a f p + c i p + g j p - a k p -
f k p)^2 +
27 (-a - f - k - p)^2 (d g j m - c h j m - d f k m +
b h k m + c f l m - b g l m - d g i n + c h i n +
d e k n - a h k n - c e l n + a g l n + d f i o -
b h i o - d e j o + a h j o + b e l o - a f l o -
c f i p + b g i p + c e j p - a g j p - b e k p +
a f k p) -
72 (-b e + a f - c i - g j + a k + f k - d m - h n -
l o + a p + f p + k p) (d g j m - c h j m -
d f k m + b h k m + c f l m - b g l m - d g i n +
c h i n + d e k n - a h k n - c e l n + a g l n +
d f i o - b h i o - d e j o + a h j o + b e l o -
a f l o - c f i p + b g i p + c e j p - a g j p -
b e k p + a f k p))^2))^(1/3)))))
If I were to add a line Simplify[%] then it doesn't return in a reasonable time on my system.
The characteristic polynomial of your matrix is a fourth-degree polynomial where the coefficients are some combination of the entries of your matrix. The eigenvalues of your matrix are the roots of that polynomial. The general formula for those roots, even with the simplified coefficients of the characteristic polynomial, is not pretty (that image is the formula for the roots of $x^4 + ax^3+bx^2+cx+d = 0$, consider what would happen if you swapped each letter there with the coefficients of the characteristic polynomial).
So it's no wonder that you get a complicated expression back, and I would much rather remember the algorithm to find eigenvalues than trying to actually memorize and use a formula.