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A compound-symmetry covariance matrix has a common diagonal entry and a common off-diagonal entry: $\Sigma =aI+cJ$. For a group of size $k$, its within-group eigenvalue is $a$ and its constant-direction eigenvalue is $a+kc$. Thus $a\ge 0$ and $a+kc\ge 0$ characterize positive semidefiniteness. When $c\ge 0$, an independent shared random intercept plus individual noise realizes this covariance. A group mean has variance $c+a/k$; a difference of two disjoint subgroup means within the same group cancels $c$.

The conduction linearization has two negative eigenvalues and a cubic factor $s^3+A{s}^2+Bs+C$, where $A=\sigma +1+\zeta$, $B=\sigma (1-r)+\sigma \zeta (1+q)+\zeta$, $C=\sigma \zeta (1+q-r)$. A double zero requires $B=C=0$, giving the displayed Bogdanov–Takens bifurcation location for $0<\zeta <1$. The remaining cubic eigenvalue is $-A$. The imaginary-pair condition $AB=C$ yields $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$ and ${\omega }^2=\sigma \zeta (1-\zeta )q/(\sigma +1)-{\zeta }^2$. Thus the oscillatory onset exists precisely for $q>q_{\ast }$ when $0<\zeta <1$.

Linearize around $A=R{e}^{iqX}$ and write $r=R^2>0$. For sideband wavenumber $l$, the sum and difference of the two amplitude sidebands, together with the field amplitude $b$, evolve underExpanding $\det (\lambda I-M_l)$ givesAt $l=0$, the eigenvalues are $-2r,0,0$; the field's constant mode must be removed if its mean is prescribed. Nonzero long-wave field modes are still dynamically active.

The partial differential equation $T_t=-\mu \Delta T-{\Delta }^{2}T+\nabla \cdot (|\nabla T{|}^2\nabla T)$ on a $2\pi$-periodic square conserves the spatial mean. On a fixed-mean subspace its first instability is at $\mu =1$ and has four real critical amplitudes. With $\mu =1+{\epsilon }^2\nu$ and $T=\epsilon (A{e}^{ix}+B{e}^{iy}+\text{complex conjugate})+\cdots$, projection onto the critical Fourier modes gives the displayed equation and its $A,B$ interchange. For $\nu >0$, squares have $|A{|}^2=|B{|}^2=\nu /5$, amplitude eigenvalues $-2\nu ,-2\nu /5$, and neutral translation phases. Rolls have a positive transverse growth rate $\nu /3$.

For a self-adjoint operator $T$, a real number $\lambda$ lies in the continuous spectrum when $T-\lambda I$ is injective with dense range but is not surjective. Such spectral values need not be eigenvalues of normalizable states. The free-particle Hamiltonian on the real line has a continuous energy spectrum.

For independent coordinate flips at rate $1/2$, a Walsh character $w_A$ is an eigenfunction with eigenvalue $-|A|$. The generator is self-adjoint and negative semidefinite. Its Dirichlet form of a Markov chain is $-\mathbb{E}[fLf]=\frac{1}{4}{\sum }_i\mathbb{E}(f({\omega }^{(i)})-f(\omega ){)}^2$.

Let $\alpha =\min \{x^{T}Qx:x\ge 0,\Vert x{\Vert }_2=1\}$ for a real symmetric matrix $Q$. Compactness gives attainment. Homogeneity shows $Q-\lambda I$ is a copositive matrix exactly when $\lambda \le \alpha$, provingIts conic program dual is the trace-normalized completely positive optimization problem. A minimizing vector gives $X=x{x}^T$, which certifies equality and dual attainment directly. In general $\alpha$ differs from the unrestricted smallest eigenvalue.

For a random vector with covariance $\Sigma$ and positive coordinate variances, let $D=\mathrm{diag}(\Sigma )$. Its correlation matrix is the covariance of $D^{-1/2}(X-\mathbb{E}X)$, so it is positive semidefinite with unit diagonal. Each entry is the correlation of the corresponding pair of variables. In particular $\mathrm{tr}(C)=p$, explaining why an eigenvalue in correlation-based principal component analysis represents that eigenvalue divided by $p$ of the total variance. The empirical version is the sample correlation matrix.

If the eigenvalues of $A$ are $\ell /2^n$ with $0\le \ell <2^n$, exact quantum phase estimation on $U=e^{2\pi iA}$ uses $n$ controlled powers $U^{2^j}$. Their Hamiltonian simulation times reach $2\pi 2^{n-1}$. Counting each supplied controlled power as one oracle query is valid in that oracle model, but does not prove a polynomial time implementation from generic sparse-matrix access. In the ordinary HHL algorithm, one instead chooses finite precision appropriate to the output error and the condition number, or exploits additional structure giving efficient access to the needed powers.

In a two-hemisphere mean-field dynamo, dipole and quadrupole subspaces can reduce to $\dot{A}=aB-A$, $\dot{B}=bA-B$, with $(a,b)=({\alpha }_0-{ϵ}_1,i\Omega -{ϵ}_2)$ and $({\alpha }_0+{ϵ}_1,i\Omega +{ϵ}_2)$ respectively. Their eigenvalues satisfy $(\lambda +1{)}^2=ab$. Strict decay is equivalent to $\mathrm{Re}(ab)+[\mathrm{Im}(ab){]}^2/4<1$. Thus dipole decay requires $F_d<1$, while quadrupole decay requires $F_q={\Omega }^2({\alpha }_0+{ϵ}_1{)}^2/4+{ϵ}_2({\alpha }_0+{ϵ}_1)<1$. Their difference is $F_q-F_d={\alpha }_0({\Omega }^2{ϵ}_1+2{ϵ}_2)$, which selects the first unstable parity when starting from a state stable in both subspaces.

For a two-dimensional positive-definite covariance matrix $\Sigma$, the centered covariance ellipse is the image of the unit circle under its positive definite square root of a matrix. Equivalently, it satisfies $x^T{\Sigma }^{-1}x=1$. Its semi-major and semi-minor axes have lengths given by the square roots of the larger and smaller eigenvalues, along the corresponding eigenvectors. The ellipse encodes a second-moment geometry; a probability-coverage interpretation needs additional information about the probability distribution and the chosen scale.

For a real random vector $X=(X_1,\dots ,X_p{)}^T$ with finite second moments, its covariance matrix collects all pairwise covariances:The diagonal entries are variances. The covariance matrix is a symmetric matrix and a positive semidefinite matrix, because for every real vector $a$,Thus one matrix describes the variance of every linear combination of the coordinates. It is a positive-definite matrix exactly when no nonzero linear combination is constant almost surely. A deterministic linear relation produces a zero eigenvalue.

For example,has unit coordinate variances, covariance $0.8$, and eigenvalues $1.8$ and $0.2$. Its unit eigenvectors $(1,1{)}^T/\sqrt{2}$ and $(1,-1{)}^T/\sqrt{2}$ describe directions of greatest and least projected variance. For a unit vector $a=(\cos \theta ,\sin \theta {)}^T$, direct multiplication gives $a^T\Sigma a=1+0.8\sin (2\theta )$: the variation in the sum direction is nine times that in the difference direction.

The spectral theorem supplies an orthonormal basis of these directions. This connects covariance matrices to principal component analysis: its leading eigenvector identifies the unit-direction projection with the largest variance. For a linear transformation $A$, the transformed random vector has covariance matrix $A\Sigma A^T$, so uncertainties propagate through a linear model by matrix multiplication.

In a convention where the CP operation squares to one on the relevant bosonic state space, a CP eigenstate has eigenvalue $+1$ or $-1$. If the Hamiltonian commutes with CP, its nondegenerate eigenstates can also be CP eigenstates. In the neutral-kaon convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, the antisymmetric flavor combination is CP even and the symmetric one CP odd. Rephasing flavor states changes the written combinations, not the invariant content.

The cross-product matrix $A\mathbf{x}=\mathbf{n}\times \mathbf{x}$ obeys $A\mathbf{n}=0$ and $A^2=\mathbf{n}{\mathbf{n}}^T-|\mathbf{n}{|}^{2}I$, by the vector triple product. On the orthogonal complement of a nonzero $\mathbf{n}$, this is $-|\mathbf{n}{|}^{2}I$; hence the two remaining complex eigenvalues are $\pm i|\mathbf{n}|$. In an orthonormal basis adapted to $\mathbf{n}$, $A$ is a planar quarter-turn multiplied by $|\mathbf{n}|$, with zero action along its axis.

For three-mode porous magnetoconvection, set $\mu =r-1-q$ and assume $0<q<\zeta /\sigma$. The transverse eigenvalues are negative. Centre-manifold invariance givesSubstitution into the first equation gives the displayed positive linear and cubic coefficients. Thus the bifurcation is a subcritical pitchfork bifurcation: the origin is stable for $\mu <0$ and unstable for $\mu >0$, while the small nonzero branches at $a^2=-{\zeta }^2\mu /(3q)$ are unstable. Simply setting $\dot{b}=0$ misses the time-derivative contribution responsible for the common factor $(1-\sigma q/\zeta {)}^{-1}$.

If a planar equilibrium with purely imaginary linear eigenvalues has radial return map $r\mapsto r+\kappa r^3+O(r^4)$, then $\kappa <0$ gives nonlinear attraction and $\kappa >0$ gives repulsion. Vanishing linear decay therefore does not imply a centre.

For normalized columns and $1\le s\le N$, the restricted isometry constant is bounded by the cumulative coherence at $s-1$. A restricted Gram matrix has diagonal one and every off-diagonal row sum at most ${\mu }_1(s-1)$. The Gershgorin circle theorem and the finite-dimensional spectral theorem bound its eigenvalues between $1-{\mu }_1(s-1)$ and $1+{\mu }_1(s-1)$. At $s=1$ the distortion is zero, and at $s=2$ it is exactly the mutual coherence.

A damped pendulum is a simple pendulum whose mechanical energy is removed by a resisting torque. Here the model is undriven, with a fixed pivot, a point mass $m>0$ at distance $\ell >0$, uniform gravitational acceleration $g>0$, and resisting torque $-bd\theta /dt$ with constant $b>0$. The angle $\theta$ is measured from the downward vertical in radians. An ideal viscous rotary dashpot realizes this torque law; tangential linear drag $-\alpha v$ on the bob gives the same equation with $b=\alpha {\ell }^2$. The moment of inertia is $m{\ell }^2$, soHere $t$ is physical time, ${\omega }_0$ is the undamped small-amplitude angular frequency, and $\gamma$ has units of inverse time. In the normalized equation $\ddot{\theta }+c\dot{\theta }+\sin \theta =0$, dots mean differentiation with respect to $\tau$, not $t$. The parameter $c$ is dimensionless. This separates the chosen damping law from the small-angle approximation: retaining $\sin \theta$ allows finite angles even when the resisting torque is linear in angular velocity.

To motivate the model, release a pendulum without a push and record how its swing shrinks. A published example by Gintautas and Hubler uses a rotary sensor built from a computer mouse to measure the frequency and damping coefficient of a physical pendulum. ^[ref] The following is a proposed experiment with a chosen viscous rotary dashpot; the plotted observations are predictions, not a reproduction of their measurements.

Mount a compact bob on a light rigid rod with a fixed pivot, couple the pivot to an adjustable viscous rotary dashpot, and measure $\ell$ from the pivot to the bob's centre. First record decay with the dashpot disconnected, then repeat with it connected at several settings; this establishes how much decay is already due to other losses. Keep the bob, rod, camera, and release angle unchanged between these trials. The simple model requires rod and damper-rotor moments of inertia to be negligible compared with $m{\ell }^2$. Otherwise measure the total moment of inertia $I$ and the distance $d$ to the assembly's centre of mass: the physical pendulum equation becomes $I{d}^2\theta /d{t}^2+bd\theta /dt+Mgd\sin \theta =0$, where $M$ is the total moving mass. A damper that also exerts an elastic restoring torque would require another term.

Place a camera perpendicular to the swing plane, include the pivot and bob centre in every frame, and determine their coordinates relative to the resting vertical. For horizontal and downward separations $x,y$, use $\theta =atan2(x,y)$; record the frame timestamps rather than assuming a frame rate. Release from rest at about $10^{\circ }$, record through several complete swings, and repeat the release. For every same-side turning point, tabulate its time $t_n$ and angular amplitude $A_n=|\theta (t_n)|$. Measure the oscillation period from differences $t_{n+1}-t_n$; comparing several periods exposes drift. Interpolating around each observed extremum reduces frame discretization error. Check that late peaks still exceed the angular resolution; taking the natural logarithm of tiny, uncertain peaks amplifies their relative errors.

In the small-angle approximation, the damped pendulum becomes a damped harmonic oscillator. For $0<\gamma <2{\omega }_0$ and release from rest at ${\theta }_0$, its exact linear-model solution isThe decaying sinusoidal envelope has coefficient $|{\theta }_0|{\omega }_0/{\omega }_d$; because release is from rest, its successive same-side extrema occur at $t_n=n{T}_d$ and have $A_n=|{\theta }_0|e^{-\gamma n{T}_d/2}$. Thus the envelope and the sequence of actual peak heights are different descriptions of the same trace. The linear damping regimes and this solution are derived in Richard Fitzpatrick, Damped Oscillatory Motion, equations 82–93.

The experiment tests two predictions independently: nearly equal same-side peak spacing and a straight line when plotting $\ln (A_n/A_0)$ against $t_n-t_0$. Its predicted slope is $-\gamma /2$, givingUse several peaks and repeated trials rather than a single ratio, and compare the frequency identity with independently measured $\ell$ and $g$. For the illustrative choices $\ell =1.00\mathrm{m}$, $g=9.81m/{\mathrm{s}}^2$, $c=0.20$, and ${\theta }_0=10^{\circ }$, the linear model gives $\gamma \approx 0.6264{\mathrm{s}}^{-1}$, $T_d\approx 2.0162\mathrm{s}$, and $A_{n+1}/A_n\approx 0.5318$. The first two positive peaks after release are therefore approximately $5.318^{\circ }$ and $2.828^{\circ }$. The undamped small-amplitude oscillation period is about $2.0061\mathrm{s}$: substantial loss per swing need not imply a large change in period.

Repeat at a larger release angle to distinguish two limitations. The relative error in replacing $\sin \theta$ by $\theta$ is about ${\theta }^2/6$ for small angles: about $0.51\%$ at $10^{\circ }$, but about $17.3\%$ at $60^{\circ }$ when compared with $\theta$. Finite-amplitude swing spacing can therefore change as the swing shrinks even if $b$ is constant. Separately, friction at the pivot or quadratic drag can invalidate the assumed linear resisting torque; exponential peak decay should be checked, not imposed on all recorded motion. Peters discusses why real pendulum damping can require nonlinear terms. ^[ref] Repeatability, residual pivot motion, changes in damper temperature, and finite camera resolution are useful checks on this proposed measurement.

The normalized equilibrium points are $(\theta ,\dot{\theta })=(k\pi ,0)$. Even multiples of $\pi$ are attracting: their linearization has eigenvalues $(-c\pm \sqrt{c^2-4})/2$, giving a stable focus for $0<c<2$, repeated negative eigenvalues at $c=2$, and a stable node for $c>2$. These are the underdamped, critically damped, and overdamped small-displacement regimes. Odd multiples of $\pi$ have eigenvalues $(-c\pm \sqrt{c^2+4})/2$ of opposite signs and are saddle equilibria. The energy dissipation of a damped pendulum explains why trajectories in the phase portrait spiral inward near a stable focus instead of closing into the recurring orbit of an undamped simple pendulum.