Exponential, logarithm and group topology

27 problems · The matrix exponential, logarithm and the topology of matrix groups.

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Ex.290Convergence of the matrix exponential and its first bounds

#matrix exponential #absolute convergence #submultiplicative norm

Ex.291The exponential of a sum: commuting and non-commuting matrices

#matrix exponential #commutation #counterexample

Exponential, logarithm and group topology

Ex.653The closure of a subgroup of GL_n is a subgroup

#topological group #closure #subgroup

Exponential, logarithm and group topology

Ex.654The derivative of the matrix exponential and the one-parameter group property

#matrix exponential #derivative #homomorphism

Exponential, logarithm and group topology

Ex.655The exponential is not injective: rotations as exponentials

#matrix exponential #rotation #kernel

Exponential, logarithm and group topology

Ex.1368A matrix of positive determinant that is not an exponential

#matrix exponential #surjectivity #commutation #diagonal matrices

Exponential, logarithm and group topology

Ex.1369Centres of GL_n and SL_n, and discrete normal subgroups

#centre #discrete subgroup #connected group

Exponential, logarithm and group topology

Ex.1370The determinant of an exponential: det e^A = e^(tr A)

#matrix exponential #determinant #trace #triangularisation

Exponential, logarithm and group topology

Ex.1371exp is a homeomorphism from symmetric to positive definite matrices

#symmetric matrices #positive definite #homeomorphism #proper map

Exponential, logarithm and group topology

Ex.1372The quotients GL_n/O(n) and SL_n/SO(n)

#homogeneous space #quotient topology #positive definite #traceless

Exponential, logarithm and group topology

Ex.1373The polar-exponential decomposition of GL_n

#polar decomposition #homeomorphism #orthogonal group #GL_n

Exponential, logarithm and group topology

Ex.1374SO(n)/SO(n-1) is the sphere

#orbit map #quotient #sphere #compact Hausdorff

Exponential, logarithm and group topology

Ex.1375The exponential maps skew-symmetric matrices onto SO(n)

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

Exponential, logarithm and group topology

Ex.1376The logarithm near the identity and local inverse of exp

#matrix logarithm #local homeomorphism #power series

Exponential, logarithm and group topology

Ex.1377Connected components of the projective linear group

#projective group #quotient #determinant #parity

Exponential, logarithm and group topology

Ex.1378Nilpotent matrices and unipotent matrices

#nilpotent #unipotent #polynomial inverse #exponential

Exponential, logarithm and group topology

Ex.1379Topology of the groups of upper triangular matrices

#triangular matrices #connected components #contractible #unipotent

Exponential, logarithm and group topology

Ex.2013A subgroup of SO(4) that is not closed

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

Exponential, logarithm and group topology

Ex.2014Compact subgroups of SL_n are conjugate into SO(n)

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

Exponential, logarithm and group topology

Ex.2015Every continuous one-parameter subgroup is e^(tA)

#one-parameter subgroup #regularity #integration trick

Exponential, logarithm and group topology

Ex.2016SO(3) is homeomorphic to the 3-sphere

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

Exponential, logarithm and group topology

Ex.2017A neighbourhood of the identity generates the connected group

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

Exponential, logarithm and group topology

Ex.2018The fundamental group of SO(3)

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

Exponential, logarithm and group topology

Ex.2019The Lie algebra of a closed subgroup is a linear subspace

#Lie algebra #closed subgroup #Lie product formula

Exponential, logarithm and group topology

Ex.2020The Lie product formula

#Lie product formula #Trotter #norm estimates #telescoping

Exponential, logarithm and group topology

Ex.2021Which matrices of the special linear group are exponentials?

#SL_2 #trace #Cayley-Hamilton #exponential image

Exponential, logarithm and group topology

Ex.2243The Dirac belt trick: a 2Ï€ rotation is non-trivial, a 4Ï€ rotation is trivial

#homotopy #lifting #SO(3) #quaternions #Dirac belt

Exponential, logarithm and group topology