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A symmetric matrix obtained from a conservative five-point discretization of a diffusion equation has row centre $-s_i/h^2$ and radius at most $s_i/h^2$, where $h>0$ is the grid spacing and $s_i$ is the sum of its four nonnegative edge coefficients. Here the matrix represents the diffusion generator, with nonnegative off-diagonal entries; boundary elimination may reduce the row radius. If every coefficient is at most $a_{\max }>0$, then $s_i\le 4a_{\max }$, and the Gershgorin bounds for symmetric matrices place every eigenvalue in $[-8a_{\max }/h^2,0]$.

For the Forward Euler method, an eigenvalue $\lambda \le 0$ has amplification factor $1+\Delta t\lambda$. Requiring $|1+\Delta t\lambda |\le 1$ for every such eigenvalue gives the sufficient absolute stability condition $\Delta t/h^2\le 1/(4a_{\max })$. Symmetry makes this also prevent growth of the Euclidean norm. For $a_{\max }=2$ and $h=0.1$, the bound is $\lambda \in [-1600,0]$ and $\Delta t\le 0.00125$. At that step size the interval maps to amplification factors in $[-1,1]$. This illustrates why cheap Gershgorin disc bounds can select a stable time step before any eigenvalues are computed.

For $|G_N^{-}⟩=(|0{⟩}^{\otimes N}-|1{⟩}^{\otimes N})/\sqrt{2}$, let $O_j$ have one Pauli X gate at site $j$ and Pauli Y gates elsewhere, and let $X_N=X^{\otimes N}$. The relations $Y|0⟩=i|1⟩$, $Y|1⟩=-i|0⟩$ show that $O_j|G_N^{-}⟩$ is proportional to $|G_N^{-}⟩$ exactly when $N$ is odd. In that case its eigenvalue is ${\lambda }_N=(-1{)}^{(N+1)/2}$, while $X_N$ always has eigenvalue $-1$.

Under the EPR criterion of reality and locality, each site's $X$ and $Y$ outcomes have predetermined values $x_j,y_j\in \{\pm 1\}$. Multiplying the $N$ certain equations $x_j{\prod }_{k\ne j}{y}_k={\lambda }_N$ gives ${\prod }_j{x}_j={\lambda }_N$ when $N$ is odd, since every $y_k$ appears an even number of times. This contradicts the certain quantum value $-1$ precisely when $N\equiv 3(\mathrm{mod}4)$. Thus the displayed GHZ state gives an infinite family of perfect-correlation contradictions with local hidden-variable theory, without an inequality.

A global hydrodynamic mode is a separated solution $a(x)e^{-i\omega t}$ satisfying the complete spatial boundary conditions of a linearized flow problem. Its eigenvalue depends on the spatial domain and variable coefficients, rather than solely on a local dispersion relation.

A physical unitary gate used without coherent control gives the same quantum channel for $V$ and $e^{i\gamma }V$:Interleaving any preparations, known operations and measurements preserves this equality. Equivalently, each fixed measurement branch with $k$ uses gains only the overall factor $e^{ik\gamma }$, which cancels in its probability. Thus unrestricted repeated uncontrolled uses do not determine the common phase of the eigenvalues. A supplied controlled unitary gate is a different resource: $|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ changes by a relative control phase when $V$ is replaced by $e^{i\gamma }V$. Such controlled access permits quantum phase estimation to measure that phase relative to the identity branch.

For a smooth invariant scalar potential, differentiating ${\partial }_{I}V(i{t}^a\varphi {)}^I=0$ at a classical vacuum gives $M^2(i{t}^a{\varphi }_0)=0$. The symmetry tangent space has dimension $\dim G-\dim H_0$, where $H_0$ is the full stabilizer subgroup. With a positive nonsingular scalar kinetic term, these are massless directions of the quadratic theory. Further zero eigenvalues are possible unless the Hessian matrix is positive on the normal complement.

For a fixed source point mass, the magnitude $g=\mu /R^2$ of the gravitational acceleration measures the local pull; the radial eigenvalue ${\lambda }_r=2\mu /R^3$ of the tidal tensor measures how that pull changes across space. Doubling $R$ divides $g$ by four but ${\lambda }_r$ by eight. For a fixed small radial separation $L$, the linear relative acceleration is $\Delta a\simeq {\lambda }_{r}L$, soA small body can therefore experience strong common gravitational acceleration with a comparatively small tidal force. A measurement of $g$ alone does not determine the tidal tensor: sources at different distances can have the same $g$ but different spatial variation.

The local stretching time scale is ${\lambda }_r^{-1/2}=\sqrt{R^3/(2\mu )}$: it marks when an initially resting separation could change by order itself if the tidal tensor stayed fixed. During free fall, $R$ changes, so this scale is not a complete trajectory or a universal disruption time. Tidal disruption additionally compares the tidal force with binding by self-gravity and material forces; $g$ by itself supplies no such comparison.

The ground-state subspace is the eigenspace of a Hamiltonian operator at its lowest eigenvalue $E_0$. A computational history state construction can have many valid quantum witnesses, hence a degenerate ground space. Perturbation bounds must optimize on this entire subspace, rather than on a chosen basis of ground states.

The linearized relative-flux matrix is $J=[(-q,q/2,0);(m,-2m,m);(f,0,-f)]$. Its characteristic polynomial has constant term $qmf<0$ for physical positive-buoyancy similarity, so at least one real eigenvalue is positive. The growing mode behaves as $z^{\lambda }$ and makes the similarity trajectory unstable with increasing height, rather than with physical time.

For the positive Laplace-Beltrami operator on a closed compact Riemannian manifold, the heat trace is $Z_M(t)={\sum }_j{e}^{-t{\lambda }_j}={\int }_M{K}_M(t,x,x)dV(x)$. It records every eigenvalue with multiplicity. In a finite free isometric quotient it equals $|U{|}^{-1}{\sum }_{u\in U}{\int }_N{K}_N(t,x,ux)dV(x)$. The orbital integrals form a class function of the ambient finite isometry group, so Gassmann equivalence gives equal quotient heat traces and hence Sunada theorem.

The rational space of level-one cusp forms has a full lattice of forms with integral Fourier coefficients. It has finite rank by the valence formula for the modular group, and is preserved by the Fourier coefficients of a composite-index Hecke operator formula. Thus every Hecke operator has a monic integral characteristic polynomial on this lattice, and its eigenvalues are algebraic integers. The common eigenvalue field is totally real by the Petersson inner product self-adjointness.

Suppose one qubit is prepared in either of the known pure states $|0⟩$ and $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, with equal prior probabilities. Measuring in the computational basis seems natural, but a tilted projective measurement gives better quantum state discrimination. The two density operators and their difference areThe Pauli matrices give ${\Delta }^2=I/2$, and the eigenvalues of $\Delta$ are ${\lambda }_{\pm }=\pm 1/\sqrt{2}$. Their absolute sum gives trace distance $D=1/\sqrt{2}$, agreeing with trace distance between qubit states.

The positive spectral projector points along the unit Bloch vector $\mathbf{n}=(-1,0,1)/\sqrt{2}$:The outcome labels $+$ and $-$ here refer to the sign of the eigenvalue of $\Delta$; they do not denote a Hadamard basis measurement. Guess $|0⟩$ after $P_{+}$ and $|+⟩$ after $P_{-}$. This sign choice is the equal-prior Helstrom measurement for two pure states, a case of the Holevo–Helstrom theorem. ^[ref]

An explicit orthonormal basis for this projective measurement iswith $P_{\pm }=|h_{\pm }⟩⟨h_{\pm }|$. Thus the real coefficient rotation angle is $\beta =-\pi /8$ relative to the computational basis, while the Bloch vector of $P_{+}$ is tilted by $\pi /4$ from the positive $z$ axis toward negative $x$. State-vector and Bloch sphere angles differ by a factor of two.

Write $a=(1+1/\sqrt{2})/2$ and $b=(1-1/\sqrt{2})/2$, so $a+b=1$. The Born rule gives the conditional outcome probabilitiesThe event $P_{+}$ has probability gap $a-b=1/\sqrt{2}=D$, attaining the variational characterization of trace distance. Both preparations are identified correctly with conditional probability $a$, so

For comparison, a $Z$ projective measurement in the computational basis has conditional outcome probabilities $(1,0)$ on ${\rho }_0$ and $(1/2,1/2)$ on ${\rho }_{+}$. Guess zero on outcome zero and plus on outcome one. The average success isand the measured total variation distance is only $1/2$. The optimal tilted projective measurement gains about $0.103553$ in absolute success probability by balancing the two conditional error rates instead of identifying $|0⟩$ perfectly at the expense of $|+⟩$.

These are statistics for alternative measurements on independently prepared copies. Applying $Z$ first and then the tilted projective measurement is a different experiment, because the first measurement in quantum mechanics changes the density operator.

For a finite-dimensional Hermitian operator $A$ with decreasing eigenvalues ${\lambda }_j$ and integer $0\le r\le d$, maximizing $\mathrm{Tr}(AB)$ over positive contractions $B$ with trace $r$ gives the sum of its leading $r$ eigenvalues. In an orthonormal eigenbasis, the diagonal entries of $B$ lie in $[0,1]$ and sum to $r$. Moving their weight to the largest eigenvalues can only increase the objective. An orthogonal projection onto their eigenvectors attains the maximum.

A Hermitian inner product is a Hermitian form $H$ on a complex vector space which has positive definiteness: $H(v,v)>0$ for every nonzero vector $v$. It is precisely a complex inner product. Only with this definiteness condition doesdefine a norm rather than a seminorm or an expression that may not be real. The standard Hermitian inner product on ${\mathbb{C}}^n$ is $H(z,w)={\sum }_j{z}_j\overline{w_j}$; positive weights $c_j$ give ${\sum }_j{c}_j{z}_j\overline{w_j}$, with squared norm ${\sum }_j{c}_j|z_j{|}^2$.^[ref]

A Hermitian inner product need not have a real matrix. For instance, with the linear-first coordinate convention,The last expression proves positive definiteness without a lengthy calculation of eigenvalues. For $z=(1,i)$ it gives $6$, while for $z=(1,-i)$ it gives $2$. These unequal values expose why a silent switch to $z^{\ast }Az$ would change this particular Hermitian inner product: with these same entries it gives $2$ at $(1,i)$. All entries must keep their roles from the matrix of a Hermitian form.

The Cauchy-Schwarz inequality for a Hermitian inner product statesEquality holds exactly when $u,v$ are linearly dependent, including when either is zero. For a nonzero $u$, outline the proof by setting $a=H(v,u)/H(u,u)$ and $r=v-au$. Then $H(r,u)=0$, and the Pythagorean theorem givesNonnegativity proves the inequality, while positive definiteness makes equality equivalent to $r=0$. The scalar $a$ has no extra conjugate because the first argument is linear.^[ref]

The Cauchy-Schwarz inequality also holds for a positive semidefinite Hermitian form, but equality need not imply linear dependence: for $H_0(z,w)=z_1\overline{w_1}$, the independent vectors $e_1$ and $e_1+e_2$ give $1=1$. Equality then describes dependence after passing to the quotient vector space by the radical of a Hermitian form. The indefinite Hermitian form in Hermitian form shows that conjugate symmetry alone does not even guarantee the inequality.

For a worked orthogonal projection, take $H_{+}(z,w)=z_1\overline{w_1}+2z_2\overline{w_2}$, $u=(1,1)$ and $v=(2,0)$. Then $a=2/3$, $au=(2/3,2/3)$ and $r=(4/3,-2/3)$, with $H_{+}(r,u)=0$ and $4=4/3+8/3$ for the squared norms. The residual is perpendicular in this Hermitian inner product even though its ordinary real dot product with $u$ is $2/3$. For complex $v=(1,i)$, the same orthogonal projection uses $a=(1+2i)/3$; rotating $v$ multiplies $a$ by $e^{i\theta }$, while rotating $u$ multiplies $a$ by $e^{-i\theta }$. This combines geometric decomposition with sensitivity to the relative complex arguments of the multipliers.

A Hermitian inner product therefore supplies more than a positive diagonal: orthogonality, orthogonal projection onto a span, and the triangle inequality for its norm make approximation and distance meaningful. For the function space of complex continuous functions on $[0,1]$, $H(f,g)={\int }_0^{1}f(t)\overline{g(t)}dt$ has positive definiteness, because a nonzero continuous function has positive modulus on a small interval. The same construction turns overlap under integration into lengths and best approximation in finite-dimensional vector subspaces.

A Hermitian matrix model replaces a spacetime field by a finite-dimensional Hermitian matrix. Its normalized integral supplies correlation functions and a zero-dimensional perturbative quantum field theory expansion. For an action built from traces of matrix powers, unitary matrix conjugation is a symmetry and the eigenvalues describe the invariant degrees of freedom. A quartic real integral needs a nonnegative highest-order coupling for convergence; a formal power series can be studied independently of this convergence issue.

A bounded operator $T:H\to H$ on a real or complex Hilbert space is Hermitian, or self-adjoint, when $T=T^{\ast }$. Equivalently,Its directional values $q_T(x)=⟨Tx,x⟩$ are real, since this identity makes $q_T(x)$ equal its complex conjugate. Reality allows either sign; a positive operator additionally requires nonnegative values in every direction.

The diagram compares Hermitian quadratic forms along paths of unit vectors. The Hermitian matrix $D=\mathrm{diag}(1,-1)$ takes the values $1$ and $-1$ on the standard basis vectors; $P=\mathrm{diag}(1,0)$ takes the values $1$ and $0$; andsatisfies $H{e}_1=e_1$ and $H{e}_3=3e_3$. These eigenvectors form an orthonormal basis, and every $x=a{e}_1+b{e}_3$ has $q_H(x)=|a{|}^2+3|b{|}^2$. Thus $H$ is a positive operator, with eigenvalues $1,3$ and operator norm $3$.

Three conditions describe different geometry. A Hermitian operator satisfies $T=T^{\ast }$; a normal operator satisfies $T^{\ast }T=T{T}^{\ast }$; a unitary operator satisfies $T^{\ast }T=T{T}^{\ast }=I$, where $I$ is the identity operator. Every Hermitian operator and every unitary operator is a normal operator. Concrete matrices separate the conditions:

The spectrum of a bounded Hermitian operator is real. In finite dimension, the spectral theorem supplies an orthonormal basis of eigenvectors, turning its action into independent real scalings. This makes the geometry easy to interpret: each eigenvalue records a signed response along an orthogonal direction. An arbitrary bounded Hermitian operator in infinite dimension need not have any eigenvectors; the general spectral theorem uses a projection-valued measure. MIT's lecture 21 develops the real spectrum and the orthogonality of eigenspaces.^[ref]

A historical connection is the study of forms with conjugate variables. In his 1901 account of Charles Hermite's work, Camille Jordan describes these forms and emphasizes their role in reducing forms with real or complex coefficients. In today's notation, $x^{\ast }Hx$ is the diagonal value of a Hermitian form; the conjugate variables make a real-valued expression possible even with complex coefficients.^[ref]

For a densely defined operator outside this bounded setting, the displayed pairing identity alone defines a symmetric operator. A self-adjoint operator must also have the same operator domain as its adjoint operator.