Topology problems

#metric #definition #axioms

#open ball #metric #definition

#open ball #interval #topology

#open sets #open intervals #definition

#closed sets #complement #examples

#open ball #Euclidean norm #definition

#open sets #definition #ball

#intermediate value theorem #characteristic polynomial #eigenvalue

#inversion #unboundedness #continuity #extension

#hypersurface #determinant #closed set

#matrix exponential #absolute convergence #submultiplicative norm

#matrix exponential #commutation #counterexample

#norms #sphere #homeomorphism #radial projection

#operator norm #singular values #trace #determinant

#Frobenius norm #inner product #trace

#Frobenius norm #operator norm #identity matrix

#Frobenius norm #max norm #equivalence of norms

#completeness #finite dimension #equivalence of norms

#matrix norms #column sum #row sum #computation

#open sets #closed sets #determinant #GL_n #SL_n

#subspaces #interior #closed sets #finite dimension

#subspaces #symmetric matrices #trace #triangular matrices

#integer matrices #discrete sets #density #rationals

#polynomial maps #continuity #determinant #adjugate

#multiplication #bilinear map #continuity #submultiplicativity

#inverse #Cramer's rule #homeomorphism #adjugate

#continuity #rank #kernel #eigenvalues #counterexamples

#topological group #determinant #homomorphism #SL_n

#connected components #general linear group #determinant #local connectedness

#connectedness #convex sets #path connected #matrices

#connectedness #singular matrices #nilpotent matrices #star-shaped

#totally disconnected #discrete subspace #integer matrices

#SO(2) #O(2) #rotations #reflections #path components

#connected components #trace #half-spaces

#Cauchy #sequences #metric

#continuity #epsilon-delta #metric

#convergence #sequences #metric

#Euclidean #Manhattan #sup norm

#norm #definition #axioms

#closed sets #complement #definition

#closed sets #closed intervals #complement

#intersections #open sets #finite

#unions #open sets #arbitrary

#boundary #open set #disjointness

#boundary #sphere #ball

#interior #closure #boundary

#limit points #sequences #closure

#unions #intersections #stability

#determinant #continuity #polynomial

#matrix convergence #entrywise convergence #sequence of matrices

#summary #matrix topology #M_n(R)

#intermediate value theorem #Perron-Frobenius #eigenvector

#convexity #Rayleigh quotient #norm #symmetric matrices

#discrete subgroup #closed set #powers of a matrix

#positive semidefinite #rank #closure

#rational matrices #density #open set

#diagonalisation #non-real eigenvalues #continuity of roots

#density #discriminant #principal minors #polynomials

#topological group #closure #subgroup

#matrix exponential #derivative #homomorphism

#matrix exponential #rotation #kernel

#low rank approximation #compactness #coercivity

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

#singular values #Weyl inequality #continuity #surjectivity

#diagonal dominance #Gershgorin #open sets #invertibility

#Rayleigh quotient #connectedness #compactness #symmetric part

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

#Frobenius norm #Cauchy-Schwarz #submultiplicativity

#operator norm #submultiplicativity #Frobenius norm

#operator norm #nilpotent #diagonal #rotation

#max norm #submultiplicativity #counterexample

#orthogonal matrices #isometry #trace cyclicity

#symmetric matrices #spectral theorem #operator norm

#orthogonal group #diameter #triangle inequality

#operator norm #ball #closure #boundary

#positive definite #cones #eigenvalues #interior

#positive semidefinite #symmetric matrices #interior

#nilpotent #idempotent #algebraic sets #nowhere dense

#invertible matrices #density #boundary #determinant

#nowhere dense #hyperplanes #open dense sets

#cones #convexity #interior #nonnegative matrices

#conjugation #composition of continuous maps #similarity

#Lipschitz #inversion #resolvent identity #operator norm

#uniform continuity #inversion #singular matrices #counterexample

#squares #determinant sign #surjectivity

#symmetric matrices #skew-symmetric #orthogonal projection #Frobenius norm

#characteristic polynomial #companion matrix #continuous section

#nilpotent matrices #fibre #Cayley-Hamilton #unbounded set

#intermediate value theorem #determinant #paths #connectedness

#hyperplane #codimension #connectedness #linear subspace

#orthogonal group #complement #singular values #path connected

#Hurwitz matrices #stability #eigenvalues #linear deformation

#determinant #level sets #SL_n #scaling

#polar decomposition #general linear group #path connected

#eigenvalues #linear deformation #path connected

#orthogonal group #components #reflections #homeomorphism

#spectral radius #star-shaped #norm ball

#nilpotent matrices #idempotents #conjugacy orbits #projections

#idempotents #projections #isolated points #trace and rank

#nilpotent matrices #star-shaped sets #path connectedness #cones

#M2(R) #rotations #operator norm #chord length

#compactness #open cover #sequential

#compact #closed #bounded #metric

#compact image #continuity #compactness

#completeness #Cauchy sequences #examples

#equivalent metrics #same topology #norms

#isometry #metric #distance-preserving

#equivalent norms #topology #finite dimension

#normed spaces #Banach #examples

#product spaces #metric #sup

#interior #closure #boundary

#connectedness #intervals #IVT

#limit points #accumulation #closure

#dense #rationals #irrationals

#compactness #Heine-Borel #closed bounded

#compact #continuous image #extrema

#connectedness #path-connectedness

#connectedness #convexity #segments

#iterated limits #continuity #subtleties

#Bolzano-Weierstrass #sequential compactness

#closed unbounded #trace #non-compactness

#completeness #Cauchy sequence #identification with R^{n^2}

#interior #boundary #GL_n

#equivalent metrics #norms on matrices #finite dimension

#submultiplicative norms #operator norm #matrix norm

#separability #dense countable subset #rational matrices

#density #determinant #polynomial identity

#exponential map #SO(3) #Rodrigues formula #homeomorphism

#projective space #projections #compact Hausdorff #homeomorphism

#orthogonal group #determinant #eigenvalue 1 #Euler's theorem

#quotient topology #conjugation #Jordan block #separation axioms

#characteristic polynomial #density #symmetric matrices

#Grassmannian #compactness #connectedness #SO(n) orbit

#orbit space #eigenvalues #Weyl chamber #quotient

#SO(3) #rotation angle #trace #normal form

#characteristic polynomial #density #similarity

#adjugate #determinant #density

#Lebesgue measure #polynomials #Fubini #induction

#Cayley-Hamilton #density #diagonalisation

#discriminant #non-real eigenvalues #closure #odd dimension

#rotations #roots of unity #density

#idempotent #projections #trace #rank

#Kronecker #pigeonhole #rotation group #dense subgroup

#spectrum #closed sets #empty interior #scaling

#LU decomposition #leading principal minors #density

#minimal polynomial #nonderogatory #density

#spectral theorem #invariant functions #eigenvalues

#special linear group #transvections #density

#rational points #circle #stereographic projection #Pythagorean triples

#Hurwitz #stability #spectrum #closure

#polynomials #density #Taylor expansion

#conjugacy orbit #closure #Jordan block

#normal matrices #closed sets #subspaces

#matrix exponential #surjectivity #commutation #diagonal matrices

#centre #discrete subgroup #connected group

#matrix exponential #determinant #trace #triangularisation

#symmetric matrices #positive definite #homeomorphism #proper map

#homogeneous space #quotient topology #positive definite #traceless

#polar decomposition #homeomorphism #orthogonal group #GL_n

#orbit map #quotient #sphere #compact Hausdorff

#orthogonal group #skew-symmetric #normal form #connectedness

#matrix logarithm #local homeomorphism #power series

#projective group #quotient #determinant #parity

#nilpotent #unipotent #polynomial inverse #exponential

#triangular matrices #connected components #contractible #unipotent

#eigenvalues #Hölder continuity #Jordan block #perturbation

#inertia #symmetric matrices #Weyl #semicontinuity

#Loewner order #positive definite #inverse #square root

#positive definite #convex cone #eigenvalues #real part

#power bounded #Jordan block #closed sets

#Newton identities #power sums #characteristic polynomial #continuity

#rank one #closed cone #connectedness #minors

#lower semicontinuity #minimal polynomial #centraliser #rank

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

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

#powers #convergence #open sets #closed sets

#spectrum #triangularisation #polynomial of a matrix

#spectral radius #Jordan form #powers of a matrix

#spectral norm #singular values #Rayleigh quotient

#operator norm #golden ratio #singular values

#transpose #operator norm #duality

#spectral radius #complex eigenvalue #operator norm

#Neumann series #invertibility #geometric series #Banach algebra

#invertible matrices #perturbation #Neumann series #open set

#trace #Cauchy-Schwarz #Frobenius norm

#Schur test #spectral radius #induced norms

#strict convexity #unit ball #parallelogram law #operator norm

#rank #discrete topology #metric #continuity

#linear functionals #Riesz representation #continuity

#Banach algebra #induced norm #equivalent norms

#equivalence of norms #compactness #finite dimension

#distance to singular matrices #singular values #Eckarte-Young #SVD

#inverse #blow-up #distance to singular matrices

#equivalence constants #singular values #rank one #orthogonal

#determinant #singular value decomposition #closure #boundary

#rank #minors #semicontinuity #nowhere dense

#rank #closure #SVD #semicontinuity

#real eigenvalues #closed sets #discriminant #2x2 matrices

#diagonalisable #Jordan blocks #counterexamples

#discriminant #distinct eigenvalues #open sets #characteristic polynomial

#orthogonal group #spheres #nowhere dense #Frobenius norm

#involutions #closed sets #unbounded #counterexample

#positive definite #boundary #symmetric matrices #determinant

#projections #compactness #symmetric matrices #nowhere dense

#topological groups #open subgroup #cosets #GL_n

#determinant #interior #boundary #level sets

#commutant #subspaces #scalar matrices #nowhere dense

#trace #density #rationals #open maps

#Baire category theorem #subspaces #covering

#derivative of determinant #Jacobi formula #adjugate #density

#derivative #inversion #Neumann series #Fréchet derivative

#roots of polynomials #Cauchy bound #Hölder continuity #characteristic polynomial

#spectral radius #continuity of eigenvalues #Hölder #characteristic polynomial

#rank #semicontinuity #minors #open sets

#adjugate #rank #minors #kernel

#squaring #determinant sign #open sets #GL_n^+

#determinant #open map #SVD #singular matrices

#determinant #level sets #components #SL_n

#rank #singular value decomposition #path connected

#Givens rotations #induction #special orthogonal group

#idempotent #involution #trace #rank #conjugacy orbits

#rank #singular values #determinant sign change #path connected

#special linear group #elementary matrices #transvections #row reduction

#special linear group #polar decomposition #path connected

#special orthogonal group #normal form #rotations #path connected

#M_2(R) #complex coordinates #singular values #determinant #Frobenius norm

#M_2 #conjugation #rotations #complex coordinates #invariants

#SO(n) #finite order #normal form of orthogonal matrices #density

#M_2 #eigenvalues #discriminant #complex coordinates #real spectrum

#rotations #SO(3) #density #Euler's theorem #finite order

#signature #Sylvester's law of inertia #connected components #symmetric matrices

#M_2 #distance to singular matrices #smallest singular value #operator norm

#M_2 #singular matrices #rank one #torus #cone

#M_2 #polar coordinates #homeomorphism #circle #GL_2 #SL_2

#M_2 #operator norm #extreme points #O(2) #unit ball

#spectral radius #Schur stability #Jury criterion #M2(R) #path connectedness

#positive definite matrices #convex cone #Sym2 #boundary

#M2(R) #convex hull #rotations #complex coordinates

#M2(R) #Hurwitz stability #path connectedness #trace and determinant

#M_2(R) #complex coordinates #idempotents #involutions #trace and determinant

#nearest rotation #Frobenius norm #SO(2) #M2(R) #Procrustes

#M_2(R) #nilpotent matrices #connected components #complex coordinates

#M_2(R) #complex coordinates #complex structures #connected components

#Baire #category #complete

#completion #Cauchy #dense embedding

#counterexample #compactness #infinite dimension

#Heine-Cantor #uniform continuity #compact

#accumulation #rationals #density

#Cantor set #compactness #closed

#compact sets #Heine-Borel #closed bounded

#sequential compactness #Bolzano-Weierstrass

#continuity #open preimages #topological definition

#limit superior #limit inferior #sequences

#compactness #open cover #finite subcover

#Baire #complete metric #category

#continuity #open preimages #multivariable

#homeomorphism #ball #cube

#isometry #Euclidean #rigid motion

#uniform continuity #Heine-Cantor #compactness

#closed subgroups #GL_n #SL_n #O_n #determinant

#compactness #O_n #SO_n #boundedness

#eigenvalues #continuity #roots of polynomial

#homeomorphism #matrix map #inverse map

#matrix exponential #continuity #series

#GL_n #open set #determinant continuity

#kernel #algebraic identity #Putnam

#density #block matrices #Schur complement

#Pfaffian #connected components #skew-symmetric matrices

#SO(3) #Cayley transform #projective plane #cut locus

#adjugate #density #polynomial identity

#complexification #determinant #commuting matrices

#Ky Fan #eigenvalues #orthogonal projections #convexity

#singular values #SVD #orbit space #Weyl inequality

#quotient space #proper maps #positive semidefinite #orbit space

#connectedness #symmetric matrices #spectral theorem #SO(n)

#conjugacy orbit #minimal polynomial #closed sets

#conjugacy orbit #Jordan form #scaling #closure

#nilpotent matrices #Jordan form #orbit closure

#irrational winding #torus #Kronecker #non-closed subgroup

#compact group #Haar measure #invariant inner product #conjugation

#one-parameter subgroup #regularity #integration trick

#SO(3) #quaternions #real projective space #double cover

#generating set #open subgroup #connected group #GL_n^+

#fundamental group #covering space #SO(3) #deck transformations

#Lie algebra #closed subgroup #Lie product formula

#Lie product formula #Trotter #norm estimates #telescoping

#SL_2 #trace #Cayley-Hamilton #exponential image

#trace #numerical range #orthonormal basis #induction

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

#stochastic matrices #Dobrushin #oscillation #convergence of powers

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

#stochastic matrices #compactness #Cesàro averages #Markov chains

#spectral radius #Gelfand formula #norms #powers

#Hurwitz #Lyapunov equation #stability #positive definite

#spectral radius #nonnegative matrices #Gelfand formula #monotonicity

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

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

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

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

#Hadamard inequality #QR decomposition #Gram-Schmidt #determinant

#Hadamard inequality #AM-GM #equality case #determinant

#determinant #Lipschitz #telescoping #Hadamard inequality

#extreme points #orthogonal group #singular value decomposition #convexity

#convex hull #orthogonal group #SVD #Krein-Milman

#spectral radius #Gelfand formula #symmetric matrices #powers

#spectral radius #continuity of eigenvalues #closure #boundary

#diagonalisable #interior #distinct eigenvalues #Jordan blocks

#diagonalisable #closure #triangularisation #continuity of roots

#discriminant #algebraic sets #polynomials #nowhere dense

#symmetric matrices #spectral theorem #genericity #distinct eigenvalues

#singular values #Courant-Fischer #perturbation #Lipschitz

#eigenvalues #symmetric matrices #Courant-Fischer #perturbation

#matrix square root #compactness #uniqueness #PSD cone

#polar decomposition #homeomorphism #orthogonal group #positive definite

#QR decomposition #Gram-Schmidt #Iwasawa #homeomorphism

#Cholesky decomposition #positive definite matrices #QR #homeomorphism

#orthogonal Procrustes #SVD #Frobenius norm #polar decomposition

#Cayley transform #skew-symmetric #special orthogonal group #chart

#eigenvectors #Möbius band #connectedness #no continuous section

#path lifting #SO(2) #square root #covering space

#complex structure #A squared = -I #2x2 matrices #homeomorphism

#conjugacy orbits #centraliser #group action #rotation matrix

#deformation retraction #homotopy equivalence #polar decomposition

#diagonalisable matrices #Weyl chamber #conjugation #path connected

#symmetric matrices #signature #eigenvalues #Weyl inequality

#fundamental group #deformation retraction #circle #homotopy

#2x2 matrices #discriminant #complex eigenvalues #components

#continuity of roots #ordered eigenvalues #Rouché #real-rooted matrices

#conjugacy orbits #complex structures #orientation #connected components #GL_n

#operator norm #unit sphere #M_2(R) #radial projection #complex coordinates

#nilpotent matrices #conjugacy orbits #cyclic vectors #connected components

#Banach #fixed point #contraction

#Baire #category #dense G-delta