# Minimum eigenvalues of a matrix with two parameters

I have a $$12 \times 12$$ matrix $$K$$ depending on 2 parameters, $$k$$ and $$\beta$$ ($$k=0.1,0.2,0.3,0.4$$ and $$\beta=0.1,0.2,0.3$$). The analytical expressions of its eigenvalues are too cumbersome to evaluate. Hence, I would like to evaluate numerically its minimum eigenvalue as $$k$$ and $$\beta$$ vary in their ranges, and plot the results.

How can I do this?

K={{-2 + 2/k + (
50000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), -(2/k),
1 - (25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), 1 - (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
0, 0, 0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), (25000000000 Tan[β])/(250000 +
250000 Tan[β]^2)^(3/2), 0,
0}, {-(2/k), -2 + 2/k + (
50000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
0, 0, 1 - (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
1 - (25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), 0, 0, 0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), (25000000000 Tan[β])/(250000 +
250000 Tan[β]^2)^(
3/2)}, {1 - (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
0, -(1/2) - 1/(-2 + k) + 10/Sqrt[Tan[β]^2] + (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), -(1/2) - 10/Sqrt[Tan[β]^2], 0,
1/(-2 + k), -((
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2)),
0, (25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2), 0, 0,
0}, {1 - (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
0, -(1/2) - 10/Sqrt[Tan[β]^2], -(1/2) - 1/(-2 + k) + 10/Sqrt[
Tan[β]^2] + (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
1/(-2 + k), 0, (
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2), 0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), 0, 0}, {0,
1 - (25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), 0,
1/(-2 + k), -(1/2) - 1/(-2 + k) + 10/Sqrt[Tan[β]^2] + (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(
3/2), -(1/2) - 10/Sqrt[Tan[β]^2],
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), 0, 0, (
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2),
0}, {0, 1 - (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
1/(-2 + k),
0, -(1/2) - 10/Sqrt[Tan[β]^2], -(1/2) - 1/(-2 + k) + 10/Sqrt[
Tan[β]^2] + (
25000000000 Tan[β]^2)/(250000 + 250000 Tan[β]^2)^(3/2),
0, (25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2), 0, 0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2))}, {0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), (25000000000 Tan[β])/(250000 +
250000 Tan[β]^2)^(3/2), 0,
0, -2 + 2/k + 20/Sqrt[k^2] +
50000000000/(250000 + 250000 Tan[β]^2)^(3/2), -(2/k) - 20/
Sqrt[k^2], 1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2), 0, 0}, {0,
0, 0, 0, -((
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), (25000000000 Tan[β])/(250000 +
250000 Tan[β]^2)^(
3/2), -(2/k) - 20/Sqrt[k^2], -2 + 2/k + 20/Sqrt[k^2] +
50000000000/(250000 + 250000 Tan[β]^2)^(3/2), 0, 0,
1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2)}, {-((
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2)),
0, (25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2), 0, 0, 0,
1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
0, -(1/2) + 10000/Sqrt[(1000 - 500 k)^2] - 1/(-2 + k) +
25000000000/(250000 + 250000 Tan[β]^2)^(3/2), -(1/2), 0,
1/(-2 + k) + (
10000 (-1000 + 500 k))/((1000 - 500 k) Sqrt[(1000 - 500 k)^2])}, {(
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2), 0,
0, -((25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2)), 0, 0,
1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
0, -(1/2), -(1/2) + 10000/Sqrt[(1000 - 500 k)^2] - 1/(-2 + k) +
25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
1/(-2 + k) + (
10000 (-1000 + 500 k))/((1000 - 500 k) Sqrt[(1000 - 500 k)^2]),
0}, {0, -((
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2)),
0, 0, (25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(
3/2), 0, 0, 1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
0, 1/(-2 + k) + (
10000 (-1000 + 500 k))/((1000 - 500 k) Sqrt[(1000 - 500 k)^2]), -(
1/2) - 1/(-2 + k) + (10000 (-1000 + 500 k)^2)/((1000 - 500 k)^2)^(
3/2) + 25000000000/(250000 + 250000 Tan[β]^2)^(3/2), -(1/
2)}, {0, (
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2), 0,
0, 0, -((
25000000000 Tan[β])/(250000 + 250000 Tan[β]^2)^(3/2)),
0, 1 - 25000000000/(250000 + 250000 Tan[β]^2)^(3/2),
1/(-2 + k) + (
10000 (-1000 + 500 k))/((1000 - 500 k) Sqrt[(1000 - 500 k)^2]),
0, -(1/2), -(1/2) - 1/(-2 + k) + (
10000 (-1000 + 500 k)^2)/((1000 - 500 k)^2)^(3/2) +
25000000000/(250000 + 250000 Tan[β]^2)^(3/2)}}

• Manipulate[Eigenvalues[mat /. {k -> k1, Beta -> b1}, -1], {k1, 0, 1}, {b1, 0, 1}]? Sep 14 at 13:16
• Your matrix seems to have a three-dimensional null-space. Maybe characterize it and project it out? Sep 14 at 19:03
• @Roman yes, it has 3 null eigenvalues...my question is related to the minimum non-null one Sep 15 at 7:57

DiscretePlot3D[Min@Chop@Eigenvalues[K /. h -> 500], {k, 0.1, 0.9, 0.1}, {\[Beta], Pi/18, Pi/2.5, Pi/36}, PlotTheme -> "Detailed"]