Companion matrix characteristic polynomial and its eigenvalues

Expert intro: This is a standard but very useful fact in linear algebra: the companion matrix is built so that its characteristic polynomial reproduces the original monic polynomial. Once that identity is in place, the eigenvalues fall out immediately.

Detailed walkthrough

Let $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.$ The companion matrix is $C=\begin{pmatrix} 0&0&\cdots&0&-a_0\\ 1&0&\cdots&0&-a_1\\ 0&1&\cdots&0&-a_2\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&-a_{n-1} \end{pmatrix}.$

We compute the characteristic polynomial $p_C(x)=\det(xI-C).$ Now

x&0&\cdots&0&a_0\\ -1&x&\cdots&0&a_1\\ 0&-1&\ddots&\vdots&a_2\\ \vdots&\vdots&\ddots&x&\vdots\\ 0&0&\cdots&-1&x+a_{n-1} \end{pmatrix}.$$ A convenient way to evaluate this determinant is by expansion along the first row, or by induction on n. The matrix has the special companion form, and the determinant satisfies the recursion that exactly reproduces the coefficients of f(x). The result is $$\det(xI-C)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=f(x).$$ So the characteristic polynomial of C is precisely f(x). Since eigenvalues are the roots of the characteristic polynomial, the eigenvalues of C are exactly the roots of f(x): $$\lambda_1=z_1,\;\lambda_2=z_2,\;\ldots,\;\lambda_n=z_n,$$ counted with algebraic multiplicity. A quick structural remark: the companion matrix is one of the cleanest matrix models for a monic polynomial, which is why it appears often in numerical linear algebra and control theory. ### 💡 Pitfall guide A common slip is to say the eigenvalues are the roots of f(x) without mentioning multiplicity. For repeated roots, they appear with the same algebraic multiplicity as in the characteristic polynomial. Another easy mistake is to compute \det(C-xI) and forget that the standard characteristic polynomial here is defined as \det(xI-C). ### 🔄 Real-world variant If the polynomial is not monic, say g(x)=b_nx^n+\cdots+b_0 with b_n\neq 1, one usually first divides by b_n and forms the companion matrix for the monic polynomial g(x)/b_n. The eigenvalues do not change under that normalization, because scaling the polynomial by a nonzero constant does not change its roots. ### 🔍 Related terms companion matrix, characteristic polynomial, eigenvalues