Rank, spectrum and norms

34 problems · Semicontinuity of rank, continuity of eigenvalues, singular values and spectral radius.

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Ex.292Every norm sphere of M_n(R) is a sphere

#norms #sphere #homeomorphism #radial projection

Ex.293Trace and determinant bounded by the operator norm

#operator norm #singular values #trace #determinant

Rank, spectrum and norms

Ex.656Existence of a closest matrix of rank at most r

#low rank approximation #compactness #coercivity

Rank, spectrum and norms

Ex.657Singular matrices have measure zero

#Lebesgue measure #zero set of a polynomial #Fubini #determinant

Rank, spectrum and norms

Ex.658The singular value map is continuous and onto the ordered cone

#singular values #Weyl inequality #continuity #surjectivity

Rank, spectrum and norms

Ex.659Strictly diagonally dominant matrices are invertible

#diagonal dominance #Gershgorin #open sets #invertibility

Rank, spectrum and norms

Ex.660The numerical range of a real matrix is an interval

#Rayleigh quotient #connectedness #compactness #symmetric part

Rank, spectrum and norms

Ex.661The Moore-Penrose pseudo-inverse is discontinuous

#pseudo-inverse #discontinuity #rank #singular value decomposition

Rank, spectrum and norms

Ex.1380Eigenvalues are only Hölder continuous

#eigenvalues #Hölder continuity #Jordan block #perturbation

Rank, spectrum and norms

Ex.1381Semicontinuity of the inertia of symmetric matrices

#inertia #symmetric matrices #Weyl #semicontinuity

Rank, spectrum and norms

Ex.1382Inversion reverses the order of positive definite matrices

#Loewner order #positive definite #inverse #square root

Rank, spectrum and norms

Ex.1383Matrices with positive definite symmetric part

#positive definite #convex cone #eigenvalues #real part

Rank, spectrum and norms

Ex.1384Power-bounded matrices do not form a closed set

#power bounded #Jordan block #closed sets

Rank, spectrum and norms

Ex.1385Power sums determine the characteristic polynomial

#Newton identities #power sums #characteristic polynomial #continuity

Rank, spectrum and norms

Ex.1386The cone of rank-one matrices

#rank one #closed cone #connectedness #minors

Rank, spectrum and norms

Ex.1387Semicontinuity of the minimal polynomial degree and of the centraliser

#lower semicontinuity #minimal polynomial #centraliser #rank

Rank, spectrum and norms

Ex.1388Where is {det = 0} a manifold?

#determinant #implicit function theorem #singular point #cone #local homology

Rank, spectrum and norms

Ex.1389The discriminant of a 2×2 matrix and a cone

#discriminant #2x2 matrices #convex cone #complex eigenvalues

Rank, spectrum and norms

Ex.1390The set of matrices whose powers converge is neither open nor closed

#powers #convergence #open sets #closed sets

Rank, spectrum and norms

Ex.1391The spectral mapping theorem for polynomials

#spectrum #triangularisation #polynomial of a matrix

Rank, spectrum and norms

Ex.1392A^k converges to 0 if and only if the spectral radius is less than 1

#spectral radius #Jordan form #powers of a matrix

Rank, spectrum and norms

Ex.2022A traceless matrix has a zero diagonal in some orthonormal basis

#trace #numerical range #orthonormal basis #induction

Rank, spectrum and norms

Ex.2023The fibres of (trace, determinant) on 2×2 matrices

#conjugacy classes #2x2 matrices #hyperboloid #trace #determinant

Rank, spectrum and norms

Ex.2024Dobrushin's contraction for positive stochastic matrices

#stochastic matrices #Dobrushin #oscillation #convergence of powers

Rank, spectrum and norms

Ex.2025Eigenvalues of normal matrices move by at most the size of the perturbation

#normal matrices #Bauer-Fike #Weyl #unitary diagonalisation

Rank, spectrum and norms

Ex.2026A stochastic matrix has a stationary distribution

#stochastic matrices #compactness #Cesàro averages #Markov chains

Rank, spectrum and norms

Ex.2027Gelfand's spectral radius formula

#spectral radius #Gelfand formula #norms #powers

Rank, spectrum and norms

Ex.2028Lyapunov's criterion for Hurwitz matrices

#Hurwitz #Lyapunov equation #stability #positive definite

Rank, spectrum and norms

Ex.2029The spectral radius is monotone on nonnegative matrices

#spectral radius #nonnegative matrices #Gelfand formula #monotonicity

Rank, spectrum and norms

Ex.2030Sylvester's criterion and the openness of the positive cone

#Sylvester criterion #positive definite #LDL decomposition #leading minors

Rank, spectrum and norms

Ex.2031The Bauer-Fike theorem

#Bauer-Fike #perturbation of eigenvalues #condition number #diagonalisable

Rank, spectrum and norms

Ex.2032The Eckart-Young theorem in the operator norm

#Eckart-Young #low rank approximation #singular values #dimension counting

Rank, spectrum and norms

Ex.2033Matrices with two distinct real eigenvalues retract onto a circle

#homotopy equivalence #deformation retraction #2x2 matrices #discriminant

Rank, spectrum and norms

Ex.2244The square root is operator monotone

#Loewner order #operator monotone #square root #positive semidefinite

Rank, spectrum and norms