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Calculate the characteristic polynomial of $\begin{bmatrix}0 & 0 & 0 & ... & 0 &a_0 \\-1 & 0 & 0 & ... & 0 & a_1 \\ 0 & -1 & 0 & ... & 0 & a_2 \\ ... & ... & ... & ... & ... & ...\\ 0 & 0 & 0 & ... & -1 & a_{n-1}\end{bmatrix}$

So we need to calculate the determinant of $\begin{bmatrix}-\lambda & 0 & 0 & ... & 0 &a_0 \\-1 & -\lambda & 0 & ... & 0 & a_1 \\ 0 & -1 & -\lambda & ... & 0 & a_2 \\ ... & ... & ... & ... & ... & ...\\ 0 & 0 & 0 & ... & -1 & a_{n-1}-\lambda\end{bmatrix}$

I solved $det(\begin{bmatrix}-\lambda & a_0 \\ -1 & a_1-\lambda\end{bmatrix})=(-\lambda (a_1 - \lambda))+a_0=\lambda^2-a_1\lambda+a_0$.

This leads me to believe that the determinant I am trying to find is: $\lambda^n-a_{n-1}\lambda^{n-1}+a_{n-2}\lambda^{n-2}-...+(-1)^na_0$, but I am struggling to prove this theory.

Gerry Myerson
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AndroidFish
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1 Answers1

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Hint: You've done the base case, so follow up with an induction.

Another hint: It would be handy that the previous case appear as a minor in expansion by minors.

For instance, \begin{align} &\det \begin{bmatrix} -\lambda & 0 & 0 & ... & 0 &a_0 \\ -1 & -\lambda & 0 & ... & 0 & a_1 \\ 0 & -1 & -\lambda & ... & 0 & a_2 \\ ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & -\lambda & a_{n-2} \\ 0 & 0 & 0 & ... & -1 & a_{n-1}-\lambda \end{bmatrix} \\ &= (- \lambda) \det \begin{bmatrix} -\lambda & 0 & 0 & ... & 0 &a_1 \\ -1 & -\lambda & 0 & ... & 0 & a_2 \\ 0 & -1 & -\lambda & ... & 0 & a_3 \\ ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & a_{n-3} \\ 0 & 0 & 0 & ... & -1 & a_{n-1}-\lambda \end{bmatrix} - (-1)\det \begin{bmatrix} 0 & 0 & 0 & ... & 0 &a_0 \\ -1 & -\lambda & 0 & ... & 0 & a_2 \\ 0 & -1 & -\lambda & ... & 0 & a_3 \\ ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & -\lambda & a_{n-2} \\ 0 & 0 & 0 & ... & -1 & a_{n-1}-\lambda \end{bmatrix} \text{.} \end{align} The first determinant (the one multiplied by $-\lambda$ ) evaluates via your inductive hypothesis and the last determinant is easy by expanding along the first row (leaving $a_0$ times a determinant of an upper-triangular matrix containing only $-1$ s and $-\lambda$ s).

Eric Towers
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