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For any complex $\lambda$, the shifted linear operator $T=\alpha -\lambda I$ is normal: expansion of $T^{\ast }T-T{T}^{\ast }$ leaves just ${\alpha }^{\ast }\alpha -\alpha {\alpha }^{\ast }=0$. Apply part (i) to this shifted normal operator:Hence a nonzero vector is an eigenvector of $\alpha$ with eigenvalue $\lambda$ if and only if it is an eigenvector of ${\alpha }^{\ast }$ with eigenvalue $\bar{\lambda }$. The eigenspaces coincide, with conjugate eigenvalues.

Proceed by induction on $\dim V$. The zero-dimensional case has the empty orthonormal basis. In positive dimension, the fundamental theorem of algebra gives a root of the characteristic polynomial, hence a unit eigenvector $e$ of $\alpha$, say with eigenvalue $\lambda$. Part (ii) makes it an eigenvector of ${\alpha }^{\ast }$ as well. Thus the line spanned by $e$ and its orthogonal complement are invariant under both operators by the introductory adjoint argument.

On $\{e{\}}^{\perp }$, the restriction of ${\alpha }^{\ast }$ is the adjoint of the restriction of $\alpha$, and the restrictions still commute. The restricted linear operator is therefore normal. By induction it has an orthonormal basis of eigenvectors. Adjoining $e$ proves an orthonormal eigenbasis exists for $V$, establishing the finite-dimensional spectral theorem by construction.

For the final deduction, use this orthonormal basis $e_j$ and write $\alpha e_j={\lambda }_j{e}_j$. DefineThe real nonnegative diagonal entries make $\beta$ a Hermitian operator, and the modulus-one diagonal entries make $\gamma$ a unitary operator. Both are diagonal in the same orthonormal basis, so they commute and $\beta \gamma e_j={\lambda }_j{e}_j$ even when ${\lambda }_j=0$. This proves the required factorization without assuming invertibility.

Conversely, if $\alpha =\beta \gamma$ with ${\beta }^{\ast }=\beta$, ${\gamma }^{\ast }\gamma =\gamma {\gamma }^{\ast }=I$, and $\beta \gamma =\gamma \beta$, then ${\alpha }^{\ast }={\gamma }^{\ast }\beta$ andThus $\alpha$ is normal. Normality is equivalent to a commuting Hermitian–unitary factorization, with the Hermitian factor even selectable nonnegative. This is the polar decomposition with a unitary extension on the kernel.

Let $G$ be a finite undirected $d$-regular graph on $N$ vertices, with adjacency matrix $A$ and normalized random walk matrix $P=A/d$. Define edge expansion and normalized conductance byAn expander family has orders tending to infinity, uniformly bounded maximum degree, and a uniform positive lower bound on $h$. For a regular graph the constant vector is an eigenvector of $P$ with eigenvalue one. Write its eigenvalues as $1={\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_N\ge -1$, and let $\delta =1-{\mu }_2$. This spectral gap measures the energetic cost of departing from a constant function.

For every mean-zero vector $f$, the spectral theorem and the graph's energy identity giveApply this to $f=1_S-|S|1/N$. Its norm squared is $|S|(1-|S|/N)$ and its energy is $e(S,S^c)/d$. ThereforeThus a uniform spectral gap gives uniform edge expansion when $d$ is fixed. This also bounds exterior vertex expansion: each exterior vertex accounts for at most $d$ boundary edges, so $|N(S)\setminus S|\ge e(S,S^c)/d$.

For the converse, take a mean-zero eigenvector $f$ for ${\mu }_2$ and choose its sign so that its positive support has at most $N/2$ vertices. Put $g=f^{+}$, which is nonzero. Truncation decreases energy in the following precise sense: for any two real numbers $a,b$,Summing over edges yields $⟨g,(I-P)g⟩\le ⟨g,(I-P)f⟩=\delta \Vert g{\Vert }_2^2$. For the threshold sets $S_t=\{v:g(v{)}^2>t\}$, every nonempty set lies in that small positive support. Integrating its boundary count gives the layer-cake identityThe Cauchy-Schwarz inequality bounds the left side bybecause the first sum is at most $d\delta \Vert g{\Vert }_2^2$ and the second at most $2d\Vert g{\Vert }_2^2$. Cancelling proves the Cheeger inequality in both directions:Hence bounded-degree regular families have uniform edge expansion exactly when their second-largest normalized eigenvalue stays uniformly below one.

Absolute eigenvalues give additional information. Put $\rho ={\max }_{i\ge 2}|{\mu }_i|$. Centering the indicators of $S,T$ and applying the operator-norm bound on the mean-zero subspace proves the expander mixing lemma:Here $e(S,T)=1_S^{T}A{1}_T$, so internal edges are counted as oriented incidences. A small $\rho$ makes edge counts resemble uniformly distributed edges. It also gives rapid random walk mixing: starting at a vertex, $P^t$ contracts the mean-zero component of its point mass by at most ${\rho }^t$ in Euclidean norm; Cauchy-Schwarz inequality then gives total variation distance at most $\frac{1}{2}\sqrt{N}{\rho }^t$ from the uniform distribution. A bipartite graph has eigenvalue $-1$ and its non-lazy walk alternates sides, even if it has good edge expansion. The lazy walk $(I+P)/2$ has nonconstant eigenvalues between zero and $1-\delta /2$, resolving this periodicity. It is important to distinguish the gap $1-{\mu }_2$ controlling cuts from the absolute bound controlling non-lazy mixing.

Here is a fully specified deterministic construction of an infinite family. Its degree is large but fixed. SetChoose the lexicographically first ordered list $a_1,\dots ,a_d\in W$ satisfyingThis finite deterministic search terminates. To prove existence, choose the list uniformly and independently. For each nonzero $u$, its signs are independent unbiased signs; the exponential-moment estimate $\cosh t\le e^{t^2/2}$ and Markov inequality give an exceptional probability at most $2e^{-d/200}$. A union bound gives total failure probability at mostThus at least one qualifying list exists. Fix it once and for all.

Let $H$ be the Cayley graph on $W$ whose $j$th neighbor of $x$ is $x+a_j$. It is an undirected $d$-regular port-labeled graph, allowing repeated edges and loops in this intermediate construction. Its characters $x\mapsto (-1{)}^{u\cdot x}$ form an orthogonal eigenbasis, with the displayed averages as normalized eigenvalues. Therefore $\rho (H)\le 1/10$. Each loop counts as one transition port; degrees here mean row sums of the adjacency matrix.

We next define the zig-zag product and prove the bound used for iteration. A port labeling of a $D_0$-regular undirected graph $F$ gives an involution ${\mathrm{Rot}}_F(v,a)=(w,b)$, recording the endpoint and reverse port. If a $d$-regular graph $H_0$ has vertex set $[D_0]$, the product has vertices $(v,a)\in V(F)\times [D_0]$. For each ordered pair of small-graph ports, move from $a$ to $a^{\prime }$ in $H_0$, follow the big edge ${\mathrm{Rot}}_F(v,a^{\prime })=(w,b^{\prime })$, and move from $b^{\prime }$ to $b$ in $H_0$. The endpoint is $(w,b)$; reversing the steps reverses the two small-graph port labels. The product is undirected, has $|V(F)|D_0$ vertices, and has degree $d^2$.

Write $B=I\otimes P_{H_0}$ and let $R$ be the permutation matrix of ${\mathrm{Rot}}_F$. The normalized product operator is $BRB$. For a mean-zero unit vector, decompose $f=u+v$ into its cloud-constant part and its cloud-orthogonal part. Then $Bu=u$, $\Vert Bv\Vert \le \rho (H_0)\Vert v\Vert$, and $\Vert R\Vert =1$. The compression of $R$ to cloud-constant vectors is exactly $P_F$, so $|⟨u,Ru⟩|\le \rho (F)\Vert u{\Vert }^2$. Expanding the quadratic form and applying Cauchy-Schwarz inequality givesThe operator is symmetric, so its mean-zero operator norm is bounded by this same quantity. No unproved product spectral estimate is needed.

Start with the loop-inclusive complete graph $G_0$ on $D$ vertices, whose normalized matrix is $J/D$ and whose nonconstant eigenvalues are all zero. Given a $D$-regular $G_t$, its two-step graph $G_t^2$ has degree $D^2$ and normalized operator $P_{G_t}^2$. Label a two-step edge by its ordered pair of ports; reversal reverses the path and its reverse ports. These $D^2$ labels are the vertex set of the fixed graph $H$. DefineIt has degree $d^2=D$, and its number of vertices is $N_{t+1}=N_t{D}^2$, hence $N_t=D^{2t+1}$. By the proved bound,Starting from zero, induction gives $\rho (G_t)\le 1/2$, since $(1/2{)}^2+21/100=46/100<1/2$. The cut-energy argument therefore gives multiplicity-counted edge expansion at least $D/4$ for every $G_t$.

To obtain ordinary simple graphs, delete loops and merge parallel edges. The maximum degree is at most $D$, and any one unordered vertex pair had multiplicity at most $D$. Thus every cut retains at least $1/D$ of its multiplicity-counted boundary, givingThese simple graphs form an expander family. Every choice is deterministic: the finite base search is fixed independently of $t$, and the recursive edge rules determine all subsequent adjacency lists. Once that base is fixed, adjacency lists can be generated successively using the previous lists, with a constant amount of work per output edge. This is squaring and zig-zag iteration yields bounded-degree expanders. The spectral argument supplies the uniform expansion guarantee and explains why squaring improves the eigenvalues while the product restores a constant degree.

The calculation is principal component analysis on a correlation matrix, rather than on an unscaled covariance matrix. For each of the eleven rating variables, subtract its column mean and divide by its column standard deviation. If $Z$ is the resulting $43\times 11$ matrix, use the divisor convention of the fitted object to writeHere the scaling standard deviations use divisor $43$, consistently with this convention. Using sample standard deviations with divisor $42$ gives the same correlation matrix and directions, with the corresponding score scale. Every diagonal entry of $R$ is one, so ${\sum }_k{\lambda }_k=\mathrm{tr}R=11$.

A principal component is the linear combination of the standardized ratings along $a_k$. Its principal component score for judge $i$ is $t_{ik}=z_i^T{a}_k$, and its variance is ${\lambda }_k$. The displayed component standard deviations are $\sqrt{{\lambda }_k}$; the explained variance of a principal component is ${\lambda }_k/11$. In particular,The first component explains about $92.122\%$ of the standardized variation, and the first two jointly explain $95.982\%$. Components three, four and five add about $2.321\%$, $0.830\%$ and $0.339\%$. The cumulative line adds these proportions successively; it is not the proportion explained by that component alone. Retaining two components discards about $4.018\%$ of the total standardized squared variation.

The displayed principal component loadings are the unit eigenvector coefficients $a_{jk}$. The first direction has positive, broadly similar weights for all ratings. It measures an overall favorable-rating tendency: a judge high on most ratings has a high first score. The second direction compares integrity and demeanour, with large positive coefficients, against such attributes as decision promptness, case-flow management and physical ability, with negative coefficients. Thus among similarly rated judges, a positive second score indicates relatively stronger integrity/demeanour compared with those operational ratings. This is a description of a data direction, not a causal interpretation. Multiplying a whole eigenvector and its scores by minus one reflects the display but changes nothing substantive.

These principal component loadings are not themselves variable-component correlations. With standardized variables,For example, the integrity correlation with component one is approximately $3.1833\times 0.289\simeq 0.920$. Blank loadings in the printed second column indicate suppressed small entries, not exact zeros. The complete unrounded loading vectors in the fitted object must be used to calculate scores.

A two-dimensional representation plots the 43 pairswith each point labeled by its judge. In the fitted software object these are the first two columns of the score matrix. Nearby points have similar projections of their standardized ratings. The rank-two reconstruction ${\hat{z}}_i=t_{i1}{a}_1+t_{i2}{a}_2$ is the orthogonal projection onto the leading principal plane and minimizes the total squared reconstruction error over all two-dimensional linear subspaces. To see this, for any orthonormal pair of directions the captured variance is the sum of its two Rayleigh quotients, at most ${\lambda }_1+{\lambda }_2$ by diagonalizing $R$; the remaining variance is the total minus that captured sum. Distances in the plot omit the smaller components, so a close projected pair need not be equally close in every original rating. The actual 43 coordinates cannot be recovered from the summary alone; the supplied data matrix or fitted score matrix is needed.

The printed descriptive list omits INTG, although the loading table includes it. INTG denotes integrity and supplies the eleventh variable. The standard data set also contains a contacts variable; the stated eleven-rating analysis uses the ratings, not contacts.

Differentiate the material derivative of the passive scalar component by component. With ${\sigma }_{ij}={\partial }_j{u}_i$,The chain rule along a Lagrangian trajectory $\dot{X}=u(X,t)$ therefore gives the covector evolution of a material scalar gradientThe transpose matters: a gradient is transported as a covector, rather than as a material separation vector. When $u_3=0$, the third row of $\sigma$ vanishes, andLet $k_h=(k_1,k_2{)}^T$ and setFor $c>0$, the matrix exponential gives the exact horizontal solutionThe two projections are onto the eigenspaces with eigenvalues $-c$ and $+c$. If $p_{-}\ne 0$, write $p_{-}=k_0(\cos \varphi ,\sin \varphi {)}^T$ with $k_0=|p_{-}|>0$. ThenThe stated growing asymptotic requires this nonzero projection. An initial gradient entirely in the other eigenspace decays, and when $c=0$ the horizontal gradient is constant.

For constant vertical shear, write $h=({\sigma }_{13},{\sigma }_{23}{)}^T$. Integrating the third component using the exact horizontal solution yieldsConsequently, if $D=h\cdot (\cos \varphi ,\sin \varphi )$ is nonzero, $k_3\sim -k_{0}D{e}^{ct}/c$, andIf $D=0$, exponential growth of $k_3$ is absent, but the ratio still tends to zero on the growing horizontal branch. Thus exponential vertical growth is generic, not unconditional. The scalar-gradient inclination in a steady planar strain is the tangent of the angle between the scalar's normal and the horizontal plane. Equivalently, it is the cotangent of the inclination of an isoconcentration surface to that plane. Large $\alpha$ corresponds to nearly horizontal scalar surfaces; small $\alpha$ corresponds to nearly vertical ones.

For the probability calculation, take $\Gamma ,\Lambda >0$. The independent random variables $a,b$ with normal distributions of variance ${\Gamma }^2$ make $c$ distributed according to the Rayleigh distribution, withThe advertised Gaussian-over-Rayleigh inclination distribution also needs $D$ to be independent of $(a,b)$; its Gaussian marginal alone does not imply this. Under that intended assumption, for $A>0$,The cumulative distribution function is zero for $A\le 0$. Differentiating givesand the probability density function is zero for negative $A$. Its integral is one because the cumulative distribution function tends to one at infinity. Independence is a real hypothesis: $D=(\Lambda /\Gamma )a$ has exactly the stated Gaussian marginal but makes $\alpha \le \Lambda /\Gamma$ almost surely, contradicting the unbounded support of this density. A sufficient physical model is isotropic Gaussian vertical shear, independent of $a,b$, whose projection in any horizontal direction has variance ${\Lambda }^2$.

If the strain varies slowly, the horizontal gradient must align with the current growing direction, and the vertical-to-horizontal ratio must adjust before that strain changes. Both adjustment times are of order $c^{-1}$. Thus the local condition is $c{\tau }_{\sigma }\gg 1$, and the typical condition for the specified statistics isThis is an adiabatic approximation, not a uniform statement about arbitrarily weak instantaneous strains: samples with $c{\tau }_{\sigma }\lesssim 1$ retain memory of earlier strain. The instantaneous eigenvector must likewise turn slowly compared with $c$.

For rapidly fluctuating vertical shear, insert the large-time horizontal solution into its stochastic differential equation:The linear combination $W=\cos \varphi W^{(1)}+\sin \varphi W^{(2)}$ is a Brownian motion: it has independent Gaussian increments and increment variance $({\cos }^2\varphi +{\sin }^2\varphi )dt=dt$. For the signed inclination $\beta =k_3{e}^{-ct}/k_0$, the Itô product rule introduces no extra cross term because the exponential factor is deterministic. It follows thatThis is an Ornstein-Uhlenbeck process, explaining why white-noise vertical shear gives an Ornstein-Uhlenbeck inclination. Its Fokker-Planck equation isIn a stationary state the Fokker-Planck probability current $j=-cBp-(g^2/2)p^{\prime }$ is constant. A nonzero constant current cannot give an integrable, nonnegative density at both ends of the real line, so $j=0$. Integrating $p^{\prime }/p=-2cB/g^2$ and normalizing with the Gaussian integral givesThe stationary variance is $g^2/(2c)$. More explicitly the explicit Ornstein-Uhlenbeck solution has mean $\beta (0)e^{-ct}$ and variance $g^2(1-e^{-2ct})/(2c)$, confirming convergence to this density. Here $c>0$ and $g\ne 0$ are needed for the stated smooth stationary density; at $c=0$ there is no confinement, while $g=0$ gives a point mass at zero.

Near a spatially extended Hopf bifurcation, a critical oscillatory normal mode has a slowly varying complex amplitude. For a selected travelling-wave branch, a method of multiple scales expansion takes the formHere the distance from onset is of order ${\epsilon }^2$, $v_0$ is the critical eigenvector, and $v_g$ is the group velocity. The comoving coordinate removes the first spatial derivative of the envelope. Projection at the first resonant order gives $A_T=\rho A+D_0{A}_{XX}-G_0|A{|}^{2}A$. Spatial translation and the temporal phase of the Hopf oscillation give a constant-phase symmetry $A\mapsto e^{i\theta }A$, which permits the cubic term $|A{|}^{2}A$ but forbids a generic $A^2$ term. The coefficients are complex because the envelope has both growth and frequency detuning. For a supercritical branch with positive real diffusion, real rescalings and a uniform phase rotation give the complex Ginzburg–Landau equationA subcritical Hopf branch requires higher saturation terms instead. If spatial reflection makes opposite travelling waves critical together, two coupled amplitude equations are required initially; the scalar equation applies to a selected single-wave branch when its competing amplitude is stable.

Distant boundaries can still determine which travelling pattern is observed. They can select its phase and wavenumber, reflect a travelling disturbance, inject a competing wave, or let a growing wave packet leave the domain. In particular, amplification in a comoving frame can be convective wave-packet instability rather than absolute wave-packet instability at a fixed laboratory point. A finite-domain outcome then need not reflect intrinsic bulk modulation dynamics. Periodic boundary conditions are the appropriate idealization when studying bulk instabilities without end selection or reflected-wave forcing: they eliminate physical end layers and make the allowed envelope Fourier modes discrete. They do not assert that real distant boundaries are always negligible. Choose a periodic length compatible with the carrier, fix the phase winding, and compare perturbations within that same periodic problem. On an open physical domain the group velocity and actual boundary conditions must be restored.

The optimum carrier wavenumber corresponds to the spatially uniform envelope $A_f=e^{-icT}$, often called a flat state. Its phase is arbitrary. Write $A=e^{-icT}(1+r)e^{i\varphi }$ with real small amplitude and phase disturbances. To first order,A Fourier mode $e^{\lambda T+ikX}$ therefore has matrix and dispersion relationThe trace is negative. The amplitude mode at $k=0$ has eigenvalue $-2$, while the phase mode is neutral because constant phase shifts preserve the solution. Put $D=1+bc$. For every nonzero real $k$, the determinant is positive if $D\ge 0$, whereas $D<0$ makes it negative for $0<k^2<-2D/(1+b^2)$. ThusThis is the Benjamin-Feir stability condition. At $D=0$, nonzero modes still decay, but the leading long-wave decay is fourth order rather than diffusive. On a periodic envelope interval of length $L$, the smallest nonzero wavenumber is $2\pi /L$, so strict stability of all allowed nonzero modes requiresA sufficiently short periodic domain can therefore exclude the unstable band even when the infinite-domain Benjamin-Feir stability condition fails.

The amplitude is damped while the phase is slow. For a slowly varying phase, the leading slaved amplitude is $r=-\frac{1}{2}\{b{\varphi }_{XX}+({\varphi }_X{)}^2\}+\cdots$. Substitution in the phase equation gives the diffusion term $D{\varphi }_{XX}$ and the nonlinear frequency shift $(c-b)({\varphi }_X{)}^2$. The latter also follows directly from a constant phase gradient: an envelope with gradient $q$ has amplitude squared $1-q^2$ and frequency $c+(b-c)q^2$. Constant phase symmetry rules out undifferentiated $\varphi$; translation invariance gives constant coefficients. The leading comoving Ginzburg-Landau equation also has spatial parity, so odd linear derivatives are absent at this order. Its neutral phase eigenvalue expands asThis coefficient follows by inserting a power series for $\lambda$ in the quadratic dispersion relation. Close to $D=0$, it becomes $E=(1+b^2)/2>0$ at leading order. The long-wave phase reduction of the complex Ginzburg-Landau equation is consequentlyThe fourth derivative regularizes negative phase diffusion. Terms omitted here have higher order in the long-wave expansion. First-order advection has been removed by the group-velocity frame; higher odd derivatives in a more general travelling-wave problem are beyond this leading amplitude approximation.

Take $D=-d$ with $d>0$ small. At the instability boundary $c=-1/b$, so $g=-(1+b^2)/b$ is nonzero. Set $\xi =\sqrt{d/E}X$, $\tau =d^{2}T/E$, and $\varphi =(d/g)\Phi$. At leading order the phase dynamics become the potential Kuramoto-Sivashinsky equation,With $u=-2{\Phi }_{\xi }$ this isThe variable $u$ is a phase-gradient disturbance, not the original physical field. Periodicity of the phase fixes ${\int }_0^{P}ud\xi =0$ for the zero-winding branch; the Kuramoto-Sivashinsky equation conserves this mean. The scaled period is $P=L\sqrt{d/E}$.

At fixed period $P$, the flat phase has growth rates ${\lambda }_n=k_n^2-k_n^4$, $k_n=2\pi n/P$. Its first loss of stability occurs at $P=2\pi$. To find what replaces it, put $k=2\pi /P$ and consider $P$ just above $2\pi$. After choosing an origin by translation, expand a small steady modulation as $u=a\sin (k\xi )+b_2\sin (2k\xi )+\cdots$. Projection of $-u{u}_{\xi }$ onto the first two harmonics yieldsSince ${\lambda }_2=-12$ at onset, this harmonic is slaved to $b_2=k{a}^2/(2{\lambda }_2)+\cdots$. The fundamental amplitude obeysThe primary periodic bifurcation of the Kuramoto-Sivashinsky equation is therefore supercritical. For $0<P-2\pi \ll 1$,The radial eigenvalue of this nonzero branch is $-2{\lambda }_1+O({\lambda }_1^2)<0$; all higher harmonics remain damped and the modulation's spatial phase is neutral. Thus the small finite-amplitude modulation is stable in the sense of orbital stability, at fixed mean and fixed period, despite the flat state's instability. The two signs of $a$ represent translations of the same modulation. In the potential formulation the spatial mean of $\Phi$ advances at $⟨({\Phi }_{\xi }{)}^2⟩$: the original wave acquires a nonlinear frequency correction rather than requiring a time-independent absolute phase.

For larger $P$, further linear modes become active and the small-amplitude argument no longer determines stability. A periodic modulation $\bar{u}$ must then be tested with the operator $v_{\tau }=-v_{\xi \xi }-v_{\xi \xi \xi \xi }-{\partial }_{\xi }(\bar{u}v)$ on mean-zero periodic disturbances. Its translation mode is always neutral, and secondary oscillatory or more complicated modulation can occur. Moreover, stability against perturbations of the same period does not prove stability against disturbances of a larger period; those require a separate sideband or Floquet theory calculation. The controlled conclusion near the first threshold is the existence of a stable small modulation family, not universal stability of all periodic Kuramoto-Sivashinsky solutions.

Put $h=1/(m+1)$ and impose homogeneous Dirichlet boundary conditions, as required by the stated continuum eigenvalues. Let $T=\mathrm{tridiag}(1,-2,1)$ be the unscaled one-dimensional Dirichlet discrete Laplacian. For $1\le k\le m$, the vectors $v_j^{(k)}=\sin (jk\pi h)$ satisfyThe identity follows by adding the two neighboring sines; the values at $j=0,m+1$ vanish. These $m$ mutually orthogonal vectors form the discrete sine transform basis. Consequently their tensor products give all $m^2$ two-dimensional modes, not merely a few candidate eigenvectors.

In a compatible grid ordering the axial operator is $A_5=h^{-2}(T\otimes I+I\otimes T)$. The nine-point operator isIndeed, the added Kronecker product supplies the diagonal neighbors with weight $1/6$, subtracts $1/3$ from each axial neighbor and adds $2/3$ to the central coefficient. Thus it has exactly the coefficients of the second discretization. On $v^{(k)}\otimes v^{(l)}$ the eigenvalues areandEquivalently, $h^2{\lambda }_{9,kl}=-10/3+4[\cos (k\pi h)+\cos (l\pi h)]/3+2\cos (k\pi h)\cos (l\pi h)/3$.

Choose the centered Dirichlet discrete Laplacian in each direction. With $T=\mathrm{tridiag}(1,-2,1)$ put $A=T\otimes I$, $B=I\otimes T$ and $\mu =\Delta t/h^2$. Thus $h^{-2}A$ and $h^{-2}B$ are the physical derivative approximations; the factor $h^{-2}$ has been absorbed into the Courant number in the written update. Equivalently use the scaled derivative matrices and the time parameter $\Delta t$.

For a one-dimensional vector with endpoint values $v_0=v_{m+1}=0$,Thus $A$ and $B$ are real symmetric negative-definite matrices. An orthonormal eigenvector basis shows that all eigenvalues of each exponential lie in $(0,1]$ for $\mu \ge 0$, soIterating gives $\Vert U^n{\Vert }_2\le \Vert U^0{\Vert }_2$ for every nonnegative step parameter, a mesh-independent unconditional stability bound. Multiplying the squared norm by the grid-area weight $h^2$ leaves the same estimate. In this natural tensor-grid choice $AB=BA=T\otimes T$, so $E(\mu ;A,B)=e^{\mu (A+B)}$: the split update is even the exact semidiscrete diffusion flow. Contractivity of a product of these negative symmetric exponentials would prove stability even without commutation.

Insert a Fourier mode $U_m^n=G^n{e}^{im\theta }$. With $r=1-2\mu$ and $\eta =e^{i\theta }$, the amplification polynomial isIts two roots have product $-\eta$, of modulus one. Thus if both roots are bounded by one, both must have modulus exactly one. Set $G=e^{i\theta /2}\lambda$. The equation becomesWhen $|r|<1$, both roots have modulus one and their separation obeys $|G_{+}-G_{-}|\ge 2\sqrt{1-r^2}>0$ for every frequency. The companion matrix has bounded entries and can be diagonalized with the two eigenvectors $(G_{+},1{)}^T,(G_{-},1{)}^T$. The uniform separation bounds the inverse eigenvector matrix. Hence every matrix power is bounded uniformly in $n$ and $\theta$, and the Parseval identity proves L2 norm stability for the two starting levels on the whole grid.

If $|r|>1$, frequencies near $\theta =\pi$ give one root outside the unit circle, so the scheme is unstable. If $|r|=1$, the Nyquist polynomial is $(G-r{)}^2$, a repeated unit root. The corresponding companion matrix has a nontrivial Jordan block and solutions containing $n{r}^n$. Thus the endpoints are unstable as well. On the whole line the individual Nyquist plane wave is not square-summable, but Fourier packets localized arbitrarily close to that frequency realize the unbounded power norm by continuity. Therefore the uniform stability of a shifted two-level advection scheme criterion isIn particular $\mu =1/2$ is stably exact with compatible starts, whereas $\mu =1$ is formally exact but unstable under arbitrary perturbations. Checking only the moduli of the roots and retaining both endpoints would miss this essential distinction.

Stability of a numerical method controls how perturbations in initial data, roundoff and local residuals propagate. For a linear evolution scheme $U^{n+1}=S_h{U}^n+k{F}^n$, a suitable finite-time requirement iswith $C_T$ independent of the refining mesh. The norm should approximate the norm of the continuous problem, such as an area-weighted L2 norm. The bound need not imply decay: $C_T=e^{CT}$ is appropriate when the equation itself permits finite-time growth. For a multilevel scheme the state includes all required time levels.

The error recurrence shows why stability matters. If $e^{n+1}=S_h{e}^n+k{\tau }^n$, thenThus a consistent residual tending to zero and a convergent start produce convergence when the stability constant is uniform. Tiny defects need not remain tiny without that bound. The Lax equivalence theorem makes this precise: for a well-posed linear initial value problem and a consistent linear approximation, stability is equivalent to convergence in the compatible solution norms. It is not a theorem for arbitrary nonlinear schemes, and a high formal order does not substitute for stability.

For constant coefficients on a periodic or infinite grid, von Neumann stability analysis diagonalizes spatial translations using Fourier modes. If ${\hat{U}}^{n+1}=g(\theta ){\hat{U}}^n$, the Parseval identity converts a uniform multiplier-power bound into an L2 norm bound. The contraction condition $|g(\theta )|\le 1$ is sufficient; a more general $|g(\theta )|\le 1+Ck$ also yields a finite-time bound. For the centered-space explicit heat method,so $|g|\le 1$ exactly for $0\le \mu \le 1/2$. Larger fixed Courant numbers amplify high frequencies. The Backward Euler method gives instead $g=[1+4\mu {\sin }^2(\theta /2){]}^{-1}$ and is contractive for every $\mu \ge 0$.

For $u_t=u_x$, forward time and centered space give $g=1+i\nu \sin \theta$, with $\nu =k/h$. Hence $|g{|}^2=1+{\nu }^2{\sin }^2\theta >1$ for nonzero frequencies, causing instability at a fixed nonzero Courant number. The directionally correct upwind finite difference scheme uses the neighbor at $m+1$ and hasIt is contractive exactly for $0\le \nu \le 1$. This also has a direct maximum-norm proof: the update is a convex combination of two old values. The continuum domain of dependence must fit inside the numerical one for a local explicit hyperbolic method to converge, giving the Courant–Friedrichs–Lewy condition. That necessary geometric restriction is not by itself a sufficient stability proof.

For systems and multilevel methods, each Fourier multiplier becomes a matrix. All amplification roots must obey the appropriate root bound, but their moduli alone are insufficient. A repeated unit root in a companion matrix gives a Jordan block; for instance $J=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ has $J^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$. Even distinct roots require control of the diagonalizing matrices uniformly in frequency and mesh. The uniform power bound from separated amplification roots supplies such control for a bounded two-root family with a positive gap. Question5 illustrates both sides: strict $0<\mu <1$ gives a uniform gap, whereas the endpoints have a double unit root and are unstable despite unit moduli.

Energy estimates are especially useful when coefficients vary or boundaries prevent Fourier diagonalization. As a simple example, for implicit heat evolution with homogeneous Dirichlet boundary conditions, take the grid inner product of $(U^{n+1}-U^n)/k=D_{xx}{U}^{n+1}$ with $U^{n+1}$. Discrete integration by parts givesThe boundary term vanishes because the boundary values are zero, proving contraction. For variable positive diffusion coefficients the corresponding flux-weighted squared differences remain nonnegative, so the argument extends without a constant-coefficient Fourier symbol. With forcing or lower-order terms, a discrete energy inequality and Gronwall inequality yield a mesh-independent finite-time bound.

A related technique is eigenvalue stability analysis of a finite difference method. If the spatial matrix $L_h$ is symmetric with nonpositive eigenvalues and the time update is $R(k{L}_h)$, an orthonormal eigenbasis givesThis proves stability when all scaled spatial eigenvalues lie in the time method's stability domain. For a nonnormal spatial or amplification matrix, the eigenvalues alone do not control powers: ill-conditioned eigenvectors or Jordan terms can cause transient growth. Direct operator-norm estimates, symmetrizers or energy estimates then provide the missing information. In particular an A-stable scalar time formula does not automatically give mesh-uniform bounds for every ill-conditioned family of spatial matrices.

Boundary conditions and starting procedures are part of the discretization. Interior Fourier analysis may fail to detect a growing boundary mode, and a multilevel method can excite a parasitic mode through a poor start. One must therefore analyze the actual finite-domain operator or include its boundary terms in an energy proof. Stability is a uniform error-propagation bound; Fourier symbols, energy identities, matrix norms and spectral analysis are complementary ways of establishing that bound.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.