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2 votes

Why is $x^2+x+1$ a factor of the minimal polynomial over $\Bbb R$ just because $x^2+x+1$ is a factor of the characteristic polynomial?

The minimal polynomial over $\mathbb{R}$ and the minimal polynomial over $\mathbb{C}$ coincide for real matrices. This is because taking the real and imaginary parts of the minimal polynomial over $\...
Joshua Tilley's user avatar
2 votes

Solving $c_{n} - 2c_{n-1} - 5c_{n-2} - c_{n-3}=0$

We can determine $(c_n)_{n\geq 0}$ by deriving a generating function \begin{align*} C(z)=\sum_{n=0}^{\infty}c_nz^n \end{align*} $C(n)$ is a rational function and we can derive the denominator of $C(z)$...
Markus Scheuer's user avatar
1 vote

How to find a matrix, characteristic and minimal polynomial of a linear operator?

In this case an opportunistic approach to find the minimal polynomial directly is actually easier than computing the characteristic polynomial. The idea is to compute repeated images by $L$ of some ...
Marc van Leeuwen's user avatar
3 votes

Solving $c_{n} - 2c_{n-1} - 5c_{n-2} - c_{n-3}=0$

For $x^3-2x^2-5x-1$, the discriminant is $361 = 19^2,$ and the roots are \begin{align*} \small r_1 = 1 + 2 \cos \left( \frac{8 \pi}{19} \right) + 2 \cos \left( \frac{18 \pi}{19} \right) + ...
Will Jagy's user avatar
  • 141k
2 votes

Solving $c_{n} - 2c_{n-1} - 5c_{n-2} - c_{n-3}=0$

The characteristic polynomial approach does indeed work. With polynomial $x^3-2x^2-5x-1$, you have roots \begin{align*} r_1 &\approx -1.28514248182979 \\ r_2 &\approx -0.221876162263191 \\ r_3 ...
Comma's user avatar
  • 113

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