characteristic polynomial of companion matrix

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• The characteristic and minimal polynomial of a companion matrix

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As suggested in the comment above, expand along the first row:

$$\mathrm{det}(tI_n-A) = \mathrm{det} \begin{pmatrix} t & 0 & \cdots & 0 & a_0 \\ -1 & t & \cdots & 0 & a_1 \\ \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -1 & t+a_{n-1} \end{pmatrix} =$$

$$t \cdot \mathrm{det} \; \begin{pmatrix} t & 0 & \cdots & 0 & a_1 \\ -1 & t & \cdots & 0 & a_2 \\ \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -1 & t+a_{n-1} \end{pmatrix} + (-1)^{1+n} a_0 \cdot \mathrm{det} \begin{pmatrix} -1 & t & 0 & \cdots & 0 \\ 0 & -1 & t & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & -1 & \end{pmatrix}$$

By induction we can replace the determinant on the left by $a_1+a_2t+\cdots+a_{n-1}t^{n-2}+t^{n-1}$ and the second matrix’s determinant is the product of its diagonals (since it’s upper-triangular). The product of the diagonals is $(-1)^{n-1}$. Therefore, the determinant is $t(a_1+a_2t+\cdots+a_{n-1}t^{n-2}+t^{n-1})+(-1)^{n+1}(-1)^{n-1}a_0$. Which simplifies to $a_0+a_1t+\cdots+a_{n-1}t^{n-1}+t^n$.

[When writing this up, don’t forget the base case for the induction.]