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For the homogeneous linear differential equation $\dot{Y}=K(t)Y$ with continuous coefficient matrix of period $P>0$, the monodromy matrix of a periodic linear system is the one-period map from $Y(0)$ to $Y(P)$. If $X$ is any fundamental matrix,Its eigenvalues are the Floquet multipliers. Multipliers are nonzero, since a fundamental matrix is invertible. Changing the starting phase conjugates $N$, so the Floquet multipliers and Jordan blocks are independent of that choice.^[ref]

This turns a continuous-time problem into repeated matrix multiplication:If $Nv=mv$, the associated solution of a differential equation obeys $Y(t+P)=mY(t)$ at every time, not merely at the sampled times. In particular, $Nv=v$ gives a nonzero $P$-periodic solution, and every nonzero $P$-periodic solution arises this way. If $Nv=-v$, the state changes sign after one period and repeats after two. A unit-modulus Floquet multiplier different from $1$ need not give repetition after one period, and a generalized eigenvector in a nontrivial Jordan block need not even give bounded motion.

For the scalar example under periodic solution, $N$ is the one-by-one matrix $(e^{2\pi a})$. Numerically solving a matrix problem uses the same idea: integrate one initial value problem for each column of $X(0)=I$ through one period, then compute the eigenvalues of $X(P)$. This is useful when an explicit solution of a differential equation is unavailable: one period determines the Floquet growth rate over arbitrarily many periods.

For piecewise constant coefficients, which are also covered by the one-period construction, $N$ is the time-ordered product of the segment matrix exponentials, with the earliest segment on the right. For example, take period $P=2$ and alternateSince $K_1^2=K_2^2=0$,Replacing the coefficient by its mean would instead give $e^{K_1+K_2}=\left(\begin{array}{cc}\cosh 1& \sinh 1\\ \sinh 1& \cosh 1\end{array}\right)$ and the incorrect rate $0.5$. The matrices do not commute. The scalar mean-coefficient formula therefore cannot simply be transferred to matrix systems. Nevertheless the Liouville formula for a fundamental matrix gives the correct determinant even in the noncommuting case:In the example, this is $1=m_{+}{m}_{-}$; it fixes the product of the Floquet multipliers, rather than either multiplier separately.

In Sur les équations différentielles linéaires à coefficients périodiques (1883), Gaston Floquet calls a function satisfying $f(t+P)=mf(t)$ periodic of the second kind, and distinguishes the genuinely periodic function obtained when $m=1$. This historical terminology captures precisely the difference between repetition of the coefficient and repetition of the state.^[ref]

For $0<a(x)\le \beta$, the explicit conservative diffusion update has weights $k{a}_{m-1/2}/h^2$, $1-k(a_{m-1/2}+a_{m+1/2})/h^2$, and $k{a}_{m+1/2}/h^2$. Under the displayed restriction they are nonnegative and sum to one. With zero Dirichlet boundary conditions, each new value is a convex combination of old interior and boundary values, proving contraction in the discrete maximum norm. The spatial matrix is also symmetric with $v^T{L}_{h}v=-h^{-2}\sum a_{m+1/2}(v_{m+1}-v_m{)}^2$. This puts its eigenvalues in $[-4\beta /h^2,0]$, so the same restriction gives L2 norm contraction. A stable recurrence need not be consistent with a different proposed differential equation.

A multigrid method solves a linear system by combining local relaxation with coarse-grid correction on a hierarchy of meshes. The key is to divide the work by scale: a suitable relaxation removes rapidly oscillating approximation error, while coarse-grid correction addresses slowly varying approximation error that relaxation barely changes. A broad error hump occupies many fine-grid points but fewer coarse-grid points; it becomes easier to resolve by relaxation after enough coarsening. This complementary action is why multigrid methods are useful for large finite difference method and finite element method systems.

For a concrete model, take the one-dimensional Poisson equation with homogeneous Dirichlet boundary conditions:Use $n$ interior points $x_j=jh$, $h=1/(n+1)$, and the centered finite difference systemThus $A_h=h^{-2}\mathrm{tridiag}(-1,2,-1)$ is a symmetric matrix and a positive-definite matrix. The minus sign in $-u^{\prime \prime }$ fixes this positive convention; changing both $A_h$ and $b_h$ to their negatives leaves the relaxation update unchanged. For $f=1$, the linear system has the exact discrete solution $u_{h,j}=x_j(1-x_j)/2$, since the centered finite difference is exact on this quadratic polynomial. In general, the exact discrete solution need not equal the sampled continuous solution: solving the linear system removes algebraic approximation error, not the global discretization error.

Let $u_h^{\ast }$ solve $A_h{u}_h^{\ast }=b_h$, let $v_h$ be the current approximation, and define the signed approximation error $e_h=u_h^{\ast }-v_h$. The linear-system residual is $r_h=b_h-A_h{v}_h$, so $A_h{e}_h=r_h$. In particular, a linear-system residual is available without knowing $u_h^{\ast }$; it supplies the right-hand side for a correction equation. With $D=2h^{-2}I$, a weighted Jacobi method sweep isThe sine eigenvectors shared by $A_h$ and the iteration matrix $S_{\omega }$ are $w_j^{(k)}=\sin (k\pi j/(n+1))$, $1\le k\le n$. Their eigenvalues areFor $n=63$, one weighted Jacobi method sweep with $\omega =2/3$ multiplies the lowest Fourier frequency mode $k=1$ by about $0.999197$, but the highest Fourier frequency mode $k=63$ by about $-0.332530$. Four sweeps retain about $99.68\%$ of the first amplitude and $1.22\%$ of the second. The sign reversal of the second amplitude is harmless when its magnitude shrinks. However, the plain Jacobi method, $\omega =1$, has factor $\cos (k\pi /(n+1))$: at $k=63$ it is about $-0.998795$. A nearly alternating mode near the grid's Nyquist limit therefore persists almost as strongly as the smoothest mode. Plain Jacobi method convergence on a fixed Dirichlet boundary condition grid does not imply effective smoothing.^[ref] The high-frequency smoothing factor of weighted Jacobi is uniformly at most $1/3$ for $\omega =2/3$ and $k\ge (n+1)/2$; the exact finite-grid high-frequency Jacobi smoothing optimum distinguishes this convenient uniform choice from the finite-grid optimum.^[ref]

One two-grid cycle applies a few weighted Jacobi method sweeps, computes the linear-system residual, performs coarse-grid correction, and applies a few more sweeps. The first relaxation makes the remaining approximation error easier to represent on a coarse mesh; the last relaxation damps oscillatory components left after the transferred correction. In a geometric multigrid V-cycle, replace the expensive exact coarse solve by one recursive cycle on the coarse correction equation, starting its correction from zero. Repeat until a tiny coarsest linear system can be solved directly, then return through the levels by adding each interpolated correction and post-smoothing. The descent and return give the cycle its V shape.^[ref]

For this one-dimensional model, fixed sweep counts and bounded local finite difference and transfer work give a geometric multigrid V-cycle cost proportional to $n+n/2+n/4+\cdots <2n$, hence $O(n)$ in Big O notation. A fixed error reduction also needs a contraction factor bounded below one independently of $h$; per-cycle work alone does not establish that. Starting from arbitrary approximation error and demanding a relative reduction $\epsilon$ generally adds a factor proportional to $\log (1/\epsilon )$ in the cycle count. Full multigrid uses coarse solutions as initial guesses on successively finer meshes to target the global discretization error efficiently. The two-grid Poisson factor with weighted Jacobi illustrates favorable convergence for this model. Other partial differential equations, strong anisotropy, or poorly chosen coarse spaces and relaxation can defeat these choices; a linear-work guarantee for arbitrary linear systems does not follow.

A historical example of the multilevel viewpoint is Achi Brandt's April 1977 paper, which analyzes adaptive meshes and reports numerical experiments on linear and nonlinear boundary value problems. This is a documented development of the approach, not a claim that the paper was its first invention.^[ref] The figure above is generated from the displayed eigenvalues, with lines joining discrete nodes; it contains no experimental measurements.

A Hermitian operator on a $2^n$-dimensional Hilbert space, with $2^n$ distinct eigenvalues all belonging to $\{c/2^n:0\le c<2^n\}$, must occupy this entire grid. This follows from the pigeonhole principle: the set of possible values and the set of distinct eigenvalues have the same cardinality. Its minimum eigenvalue is zero, so the operator is necessarily singular. Distinctness therefore does not imply invertibility; it forces a one-dimensional kernel in this specific complete-grid setting.

For a positive-definite symmetric matrix $A$, the ellipsoid $x^{T}Ax=1$ has nearest points to the origin exactly at $x={\lambda }_{\max }^{-1/2}u$, where $u$ ranges over unit vectors in the largest-eigenvalue eigenspace. Indeed, the spectral theorem for real symmetric matrices gives $x^{T}Ax\le {\lambda }_{\max }|x{|}^2$, with equality precisely in that eigenspace. A simple largest eigenvalue gives two opposite nearest points; multiplicity $m>1$ gives a scaled $(m-1)$-sphere of nearest points.

A real symmetric matrix $A$ is negative definite when $x^{T}Ax<0$ for every nonzero real vector $x$. Equivalently, $-A$ is a positive-definite matrix and every eigenvalue of $A$ is negative.

A symmetric matrix $A$ is negative semidefinite when $v^{T}Av\le 0$ for every vector $v$. Equivalently, all its eigenvalues are nonpositive.

The nonnegative Neumann Laplacian is the self-adjoint realization of $-\mathrm{div}\mathrm{grad}$ with vanishing outward normal derivative. Its weak quadratic form is the Dirichlet energy on $H^1$. On a connected compact domain the constants give a one-dimensional zero eigenvalue space.

With CPT symmetry, a Hermitian kaon mass matrix is $M=\left(\begin{array}{cc}{r}_0& z\\ z^{\ast }& r_0\end{array}\right)$. Its eigenvalues are $r_0\pm |z|$. For correlated roots $a^2=z$, $b^2=z^{\ast }$ and $ab=|z|$, the normalized lower and upper eigenvectors are $(a,-b)/\sqrt{2|z|}$ and $(a,b)/\sqrt{2|z|}$. They are orthogonal. This is a mass-only result; including decay requires the generally non-Hermitian effective matrix $M-i\Gamma /2$.

A newform is a normalized Hecke eigenform in the new subspace of cusp forms. Its weight, level and Dirichlet character are part of its specification. For a newform, its Fourier coefficients equal its eigenvalues of Hecke operators.

On the new subspace of cusp forms, a common eigenspace for the Hecke operators away from the level has dimension one. Distinct newforms, even at different primitive levels, have different systems of these eigenvalues. Consequently the commuting bad-prime operators also act by scalars on these lines. This is the multiplicity-one part of Atkin-Lehner-Li theory, not a consequence of normal operator diagonalization alone. See Stein's statement of the theorem.

Newton's identities let two different summaries of the same polynomial roots talk to each other. Elementary symmetric polynomials record products of distinct root positions; power sums record sums of individual powers, $p_r={\sum }_{i=1}^n{x}_i^r$. Either can be easier to obtain in a particular problem. The identities turn one into the other, often avoiding any need to find the individual polynomial roots.

Write $e_0=1$ and $e_m=0$ for $m>n$. For every positive integer $m$, the identity isIt is a polynomial identity over the integers, so it remains true on substitution in any commutative ring. To solve it for $e_m$ by division, however, $m$ must have a multiplicative inverse. Over a field of characteristic zero, the first $n$ power sums recover every elementary symmetric polynomial. The same recovery works when the positive characteristic of a field is greater than $n$; small characteristic needs care.^[ref]

The first three relations are $e_1=p_1$, $2e_2=e_1{p}_1-p_2$, and $3e_3=e_2{p}_1-e_1{p}_2+p_3$. When division by two and three is permitted, this becomesFor the three roots $1,2,3$, their power sums are $p_1=6$, $p_2=14$, and $p_3=36$. The formulas give $e_2=(36-14)/2=11$ and $e_3=(216-252+72)/6=6$, recovering the polynomial $z^3-6z^2+11z-6$ from aggregate data.

The beauty is that the summaries do not depend on the order of the polynomial roots. In symmetric polynomials, one can work with quantities that survive every relabeling; in linear algebra, traces of matrix powers give the same kind of aggregate information about eigenvalues. The identities convert those summaries into coefficients of a characteristic polynomial.

A short derivation packages the identities into a formal power series. Let $E(t)={\prod }_i(1+x_{i}t)={\sum }_{m\ge 0}{e}_m{t}^m$. Since its constant coefficient is one, it has an inverse as a formal power series, and its formal derivative satisfiesMultiply by $E(t)$ and compare the coefficient of $t^{m-1}$. This gives Newton's identities without a convergence assumption or division by $m$. The division issue arises only when using them to recover $e_m$. The logarithmic-derivative approach is also developed in Newton's Identities (January 2019 notes) by Milan Mossé (§4, pp. 6–8).

The alternate name Newton-Girard identities recalls Albert Girard's earlier work. A modern historical account credits formulas through sixth powers to Albert Girard in 1629 and discusses their later appearance in Isaac Newton's Arithmetica universalis.^[ref] This citation refers to that account; the original historical editions have not been inspected here.

A planar equilibrium with real nonzero eigenvalues of the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Repeated real eigenvalues may give a degenerate or improper node, depending on the eigenspace.

A critical point of a real function with continuous second partial derivatives is nondegenerate when its Hessian matrix is an invertible matrix there, equivalently when none of its eigenvalues is zero. A positive-definite matrix gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and opposite-sign eigenvalues give a saddle point of a scalar function. A nondegenerate critical point need not be an extremum: $x^2-y^2$ has Hessian matrix $\mathrm{diag}(2,-2)$, which is an invertible matrix.

Such critical points are the local building blocks of Morse theory.

An equilibrium is nonhyperbolic when its Jacobian matrix has at least one eigenvalue on the imaginary axis. Linearization alone then need not determine the local dynamics, and a center manifold often supplies the relevant reduction.