Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

Depends on

• Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
• In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose
• A real normal endomorphism is orthogonally block-diagonalisable with 1x1 real blocks and 2x2 rotation-scaling blocks

Used by

• Adjacency spectrum, spectral radius, and cospectral graphs Definition
• Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface Definition
• The algebraic connectivity of a finite simple graph Definition
• An explicit real symmetric 3x3 matrix is orthogonally diagonalised Example
• Polar cartan decomposition of sl n r Example
• The non-negative square root of an explicit matrix is exhibited as a polynomial in the matrix Example
• A C1 planar gradient at a nondegenerate saddle has local stable and unstable curves Lemma
• A half-line first-order operator with a hyperbolic limit has a right inverse Lemma
• Constant vector is a top eigenvector Lemma
• KAK integration formula for K-bi-invariant functions on SL2(R) Lemma
• Mixed boundary hyperbolic passage has uniform endpoint derivative bounds Lemma
• Positive square root of a compact positive operator Lemma
• Smooth Euclidean hypersurface graphs and compact localization Lemma
• Trace riccati inequality Lemma
• For Hermitian matrices, the Rayleigh quotient and residual converge with the expected rates along power iteration Proposition
• Rigidity in bishop gromov on an interval Proposition
• Rigidity in rauch comparison Proposition
• A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S Theorem
• A symmetric second-order principal part has a coordinate-invariant signature normal form Theorem
• Cayley transforms connect theta-stable Cartans in the classification Theorem
• Conjugacy of Cartan involutions Theorem
• Conjugacy of compact real forms Theorem
• Courant-Fischer min-max principle for self-adjoint endomorphisms on finite-dimensional real inner product spaces Theorem
• Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
• Existence of a Cartan involution Theorem
• Global Cartan decomposition for a connected finite center semisimple Lie group Theorem
• Hessian comparison for distance under sectional curvature bounds Theorem
• Margulis family has uniform spectral gap Theorem
• Maximal abelian subspaces of p are conjugate by K Theorem
• Near a simple Hermitian eigenvector, Rayleigh-quotient iteration converges cubically Theorem
• Restricted root space decomposition Theorem
• Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric Theorem
• The multiplicity of the Laplacian eigenvalue 0 equals the number of connected components Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)