⚡ Pluto.jl ⚡

Problem 1. Consider the hermitian matrix:

$O = (\begin{matrix} 0 & 0 & i \\ 0 & 0 & 0 \\ - i & 0 & 0 \end{matrix})$

Find the eigenvalues of $O$ using the fact that:

$tr (O) = λ_{1} + λ_{2} + λ_{3}$

$tr (O^{2}) = λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}$

$tr (O^{3}) = λ_{1}^{3} + λ_{2}^{3} + λ_{3}^{3}$

12.2 μs

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using SymPy

124 μs

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begin
    𝑖 = sympy.I
    𝐎 = sympy.Matrix([ 0 0 𝑖;
                       0 0 0;
                      -𝑖 0 0]);
    𝚝𝚛(X) = sympy.trace(X);
end;

55.9 ms

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begin
    λᵢ = sympy.symbols("λ₁ λ₂ λ₃")
end;

147 μs

Now get the equation representing the equalities in the problem statement.

7.2 μs

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get_eq(X,λ;p=1) = sympy.Eq(sum(λ.^p),𝚝𝚛(X^p));

59.7 μs

eq1

$λ ₁ + λ ₂ + λ ₃ = 0$

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eq1 = get_eq(𝐎,λᵢ)

1.3 ms

eq2

$λ ₁^{2} + λ ₂^{2} + λ ₃^{2} = 2$

x
 
eq2 = get_eq(𝐎,λᵢ,p=2)

1.8 ms

eq3

$λ ₁^{3} + λ ₂^{3} + λ ₃^{3} = 0$

x
 
eq3 = get_eq(𝐎,λᵢ,p=3)

2.3 ms

x
 
solutions = solve([eq1,eq2,eq3],λᵢ,dict=true);

18.3 ms

48.0 μs

Various eigenvalue solutions:

${λ ₁ : - 1, λ ₂ : 0, λ ₃ : 1}$

387 μs