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Make $V$ a $k[x]$-module by $xv=\phi (v)$. The structure theorem over the Euclidean domain $k[x]$ decomposes its torsion module aswhere the $p_{\alpha }$ are monic irreducibles. Each summand is indecomposable: its submodules form a chain, so two nonzero submodules cannot form a direct sum. This is the desired decomposition into invariant indecomposable subspaces.

The characteristic polynomial iswhereas the minimal polynomial is the least common multiple

Let $B$ be the matrix whose $i$th column consists of the coordinates of ${\beta }_i$ in the basis $(e_j)$. Define the upper-triangular matrix $U$ and lower-triangular matrix $L$ byThe first identity in part b says $I=BU$, so $B=U^{-1}$. The second says that the matrix $C$ of $\sigma (c)$ is $C=BL=U^{-1}L$. Since $U$ has diagonal entries one, $\det U=1$, and thereforeThus $\det (tU-L)$ is the characteristic polynomial of the Coxeter element in its geometric representation.

For a nonzero positive-weight eigenform at a good prime, the two oldforms $f(\tau ),f(p\tau )$ are independent. At level multiplied by $p$, the Hecke operator is the bad-prime operator, with matrix $\left(\begin{array}{cc}\lambda & 1\\ -\chi (p)p^{k-1}& 0\end{array}\right)$. Distinct roots $\alpha ,\beta$ of its characteristic polynomial give eigenforms $f-\beta f(p\tau )$ and $f-\alpha f(p\tau )$. Weight-zero constants are an exception to the independence assertion.

A principal minor is the determinant of a square submatrix selected using the same index set for rows and columns. For a three-by-three matrix, the coefficient of $\lambda$ in its monic characteristic polynomial is the sum of its three principal minors of order two.

The Routh-Hurwitz criterion tests whether all roots of a real polynomial have negative real part, by sign conditions on determinants formed from its coefficients. Applied to a characteristic polynomial, it gives an algebraic test for strict linear asymptotic stability.

ConsiderIts row sums vanish, so $(1,1,1{)}^T$ is an eigenvector for the zero eigenvalue. Its characteristic polynomial iswith sum of principal minors of order twoThe other eigenvalues are $(-S\pm \sqrt{S^2-4T})/2$. No symmetry is required for this formula; positive diagonal symmetrization of a matrix is a sufficient condition for real roots.

The model matrix $B=\left(\begin{array}{cc}0& 1\\ ϵ& d\end{array}\right)$ has a defective double eigenvalue when $d=ϵ=0$. Its characteristic polynomial is ${\lambda }^2-d\lambda -ϵ$, giving the displayed roots. For fixed nonzero $d$ the small root is $-ϵ/d+O({ϵ}^2)$ and the other is $d+ϵ/d+O({ϵ}^2)$. These expansions cease to be ordered at $d=O(\sqrt{ϵ})$, the distinguished limit in which both roots split on a square-root scale. This is the generic leading behavior when a perturbation closes a two-dimensional Jordan block through a nonzero lower-left entry; it need not occur for a perturbation that leaves that entry zero.

If $A$ is diagonalizable, the anticommutator map $B\mapsto AB+BA$ has eigenvalues $2{\lambda }_1$, ${\lambda }_1+{\lambda }_2$ twice, and $2{\lambda }_2$. Their product gives the displayed characteristic polynomial. The density of diagonalizable complex matrices and continuity of polynomial coefficients prove the formula also for defective matrices.

Let $L$ be the symmetric zero-diagonal Jacobi matrix with $L_{j,j+1}=L_{j+1,j}=a_j$, and let the skew-symmetric matrix $B$ haveThen $\dot{L}=[B,L]$ is equivalent to the Finite Volterra lattice. The corresponding similarity flow makes the eigenvalues of $L$, the coefficients of its characteristic polynomial, and its trace invariants first integrals.