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With canonically normalized gauge field kinetic terms, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its eigenvalues are squared physical gauge boson masses. For a complex scalar field vacuum ${\Phi }_0$ in the Higgs mechanism, $(M^2{)}_{ab}=2\mathrm{Re}[(g_a{T}_a{\Phi }_0{)}^{†}(g_b{T}_b{\Phi }_0)]$. Thus for real $x_a$, $x^T{M}^{2}x=2\Vert {\sum }_a{x}_a{g}_a{T}_a{\Phi }_0{\Vert }^2\ge 0$: it is a Gram matrix and a positive semidefinite matrix. An unbroken generator annihilates the vacuum expectation value and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.

Writing $Y=GZ$ with an invertible matrix-valued function gives $Z^{\prime }=\Lambda Z$ exactly when $\Lambda =G^{-1}AG-G^{-1}{G}^{\prime }$. For a formal power series transformation at an irregular singular point, coefficient comparison recursively removes off-diagonal terms when leading diagonal eigenvalues are distinct. This is a differential-system change of frame, not specifically an electromagnetic gauge transformation.

Rolling a sheet into a circular cylinder bends it without changing its internal lengths and angles. Fitting that same sheet smoothly onto a sphere requires distortion of its first fundamental form. Gaussian curvature distinguishes these situations: it measures the intrinsic curvature of a smooth surface, even though its simplest definition uses how the surface bends in Euclidean space.

The Gaussian curvature at a point is the determinant of the shape operator, or the product of the two principal curvatures:Here $I=Ed{u}^2+2Fdudv+Gd{v}^2$ is the first fundamental form and $II=ed{u}^2+2fdudv+gd{v}^2$ is the second fundamental form, with $S=-dN$. Regularity makes $EG-F^2>0$. Reversing the unit normal negates $e,f,g$ and both principal curvatures, leaving Gaussian curvature unchanged. ^[ref]

The sign describes bending at the point: positive Gaussian curvature means the principal curvatures have the same nonzero sign, as at an elliptic point; negative Gaussian curvature means opposite signs, as on a saddle; zero Gaussian curvature means at least one principal curvature vanishes. Zero Gaussian curvature at a single point does not imply that a neighborhood is a plane. ^[ref]

Four worked models make the signs concrete:

The patches use a common length unit, with $R=1$ for the sphere and circular cylinder. The marked red and blue curves have principal directions at the black point; the arrow is the chosen unit normal. For physical lengths, principal curvature and mean curvature have units of inverse length, while Gaussian curvature has units of inverse squared length. Rescaling a surface by $\lambda >0$ divides its principal curvatures by $\lambda$ and its Gaussian curvature by ${\lambda }^2$. Writing the saddle as $h=(x^2-y^2)/(2L)$ restores an explicit length scale: $K(0)=-1/L^2$.

The Theorema Egregium says that Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. A local isometry therefore preserves Gaussian curvature, even if the shape operator and the principal curvatures change. ^[ref]

Here is the outline, rather than a full proof. The Gauss formula splits second derivatives of a parametric surface into tangential Christoffel symbols and the second fundamental form. The Christoffel symbols are determined by the first fundamental form and its first derivatives. Commuting the next partial derivatives gives the Gauss equation, which identifies $\det S$ with the intrinsic sectional curvature computed from the Riemann curvature tensor. Thus the apparently extrinsic product $k_1{k}_2$ is determined by the Riemannian metric. ^[ref]

One can see the distinction directly by rolling a plane strip with coordinates $(u,v)$ into the circular cylinder $X(u,v)=(R\cos (u/R),R\sin (u/R),v)$. Both have $I=d{u}^2+d{v}^2$, so the map is a local isometry, and is one-to-one on an open strip of width less than $2\pi R$. Extending $u$ over all real numbers wraps around periodically; it does not give a global isometry between the entire plane and the entire circular cylinder. A nonempty open patch of a sphere cannot be flattened by a local isometry into a plane, because its Gaussian curvature $1/R^2$ cannot become zero. The obstruction concerns preservation of the first fundamental form, rather than a particular drawing or choice of seam.

The normalized Probabilists' Hermite polynomials diagonalize the Gaussian number operator with eigenvalues $n\ge 0$. Its self-adjoint closure has domain $\sum n^2|c_n{|}^2<\infty$. Polynomial truncations are dense for its graph norm. The semigroup $e^{-tN}$ is the Ornstein-Uhlenbeck semigroup and its energy form is the Gaussian Dirichlet energy.

The generalized eigenspace of a linear operator $A$ for an eigenvalue $\lambda$ is the union of the kernels $\ker (A-\lambda I{)}^N$. In finite dimension this union stabilizes for sufficiently large $N$.

A generalized eigenvalue problem asks for a nonzero vector $v$ and scalar $\lambda$ satisfying $Av=\lambda Bv$. When $B$ is invertible, it is equivalent to the ordinary eigenvalue problem $B^{-1}Av=\lambda v$.

A generalized Hopf bifurcation has a vanishing cubic radial coefficient and nonzero quintic coefficient. The two parameters unfold the eigenvalue real part and cubic coefficient. With $R=r^2$, periodic orbits satisfy $\tau +aR+b{R}^2=0$. Their saddle-node bifurcation curve is $\tau =a^2/(4b)$ with $R=-a/(2b)>0$, hence $ab<0$. For $b>0$ the smaller-radius limit cycle is stable and the larger one unstable where both exist; their collision explains why the criticality change of a Hopf bifurcation curve brings a saddle-node bifurcation of periodic orbits.

// Destination: partial-differential-equation.bigb

Every finite-dimensional complex Lie algebra representation of a nilpotent Lie algebra decomposes into invariant generalized weight spaces. The nilpotence assumption is essential: a general solvable action can have extensions between distinct weights.

Here is a proof. For $h,k\in H$, nilpotence gives $(\mathrm{ad}h{)}^{N}k=0$, hence $(\mathrm{ad}\rho (h){)}^N\rho (k)=0$. By the nilpotent commutator preserves generalized eigenspaces lemma, every generalized eigenspace of $\rho (h)$ is invariant under every $\rho (k)$. Split successively by the acting operators of a basis of $H$. Each resulting block is invariant under $H$, and each basis operator has only one eigenvalue on it. The Lie theorem triangularizes the action on each block; its diagonal entries therefore give one and the same linear functional $\lambda$ on every basis vector of that block. Every $\rho (h)-\lambda (h)I$ is strictly upper triangular there, proving that the block is $V^{\lambda }$. Distinct blocks have distinct weights, so this is a direct sum. On the zero generalized weight space, every acting operator is a nilpotent endomorphism, so the Engel theorem supplies a nonzero common annihilated vector whenever that space is nonzero.

For a real symmetric matrix, or more generally a Hermitian matrix, the finite-dimensional spectral theorem makes all eigenvalues real. Intersecting its Gershgorin discs with the real axis givesThe union preserves gaps that the single outer interval loses. If every $a_{ii}-r_i>0$, all eigenvalues are positive, so the matrix is a positive-definite matrix. This is the positive-diagonal strictly diagonally dominant matrix criterion. For the three-by-three example, $1.7\le \lambda \le 9.4$ gives a positive-definite matrix certificate and bounds its spectral condition number of a positive-definite matrix by $9.4/1.7\approx 5.529412$; the computed value is about $4.544364$.

Strictness matters. The symmetric matrix $\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$ has positive diagonal and is a diagonally dominant matrix, but its eigenvalues are $0$ and $2$: it is singular. For a general complex matrix, the condition $\mathrm{Re}{a}_{ii}>r_i$ only puts the eigenvalues in the open right half-plane. The Hermitian matrix assumption is what converts that statement into positive definiteness.

The Gershgorin circle theorem locates every eigenvalue of a square matrix using only its entries. A diagonal entry supplies a centre in the complex plane, and the sum of the off-diagonal absolute values in its row supplies a radius. Small off-diagonal couplings therefore confine the eigenvalues near the diagonal entries. This turns a global eigenvalue problem into a picture built one row at a time.

Precisely, let $A=(a_{ij})\in {\mathbb{C}}^{n\times n}$, with $n\ge 1$, and define the closed Gershgorin discsEvery eigenvalue $\lambda$ of $A$ satisfies $\lambda \in {\bigcup }_i{D}_i(A)$, including eigenvalues on the boundaries. No symmetry or diagonalizability is required. Despite the name, the theorem uses filled disks, not just their circular boundaries.^[ref]

The short reason is the triangle inequality. For a nonzero eigenvector $x$ with $Ax=\lambda x$, choose $i$ with $|x_i|={\max }_j|x_j|>0$. The $i$th row givesThus the largest eigenvector coordinate identifies a Gershgorin disc containing $\lambda$, even though that eigenvector need never be computed to use the bound.

This is useful when a full eigenvalue computation would be expensive: forming the Gershgorin discs needs one pass over the nonzero entries of a sparse matrix. If $0$ lies outside every Gershgorin disc, then $A$ is an invertible matrix, since zero is not an eigenvalue. The condition is exactly $|a_{ii}|>r_i$ in every row, namely strict diagonal dominance. This connects diagonal dominance to solvability of a linear system; the inverse infinity-norm bound from diagonal dominance adds a quantitative bound. It is a sufficient test: failure to exclude zero does not prove singularity.

For a concrete symmetric matrix, takeThe three Gershgorin discs have centres $2,5,9$ and are pairwise disjoint. The finite-dimensional spectral theorem makes the eigenvalues real, so their possible locations reduce to the intervals $[1.7,2.3]$, $[4.5,5.5]$, and $[8.6,9.4]$. Gershgorin eigenvalue counting gives exactly one eigenvalue in each. Independently, the characteristic polynomial iswhose numerical polynomial roots are $1.985811$, $4.989942$, and $9.024247$, rounded to six decimal places. All are positive, as Gershgorin bounds for symmetric matrices already guarantee without finding those polynomial roots.

S. Gerschgorin published the result in 1931 in Über die Abgrenzung der Eigenwerte einer Matrix, in the Bulletin de l'Académie des Sciences de l'URSS, issue 6, pages 749–754. The original paper's §2 states both eigenvalue inclusion and counting in separated groups of Gershgorin discs; §3 already discusses column bounds and transformations preserving the eigenvalues. Thus both the localization picture and ways to sharpen it were part of the original development.^[ref]