Maths WorkshopMaths problems with full solutions

Connectedness

30 problems · Path-connected sets of matrices, components of GLₙ and related groups.

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Ex.308Components of the linear group

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

Ex.309Convex sets of matrices are path connected

#connectedness #convex sets #path connected #matrices

Ex.310Singular matrices and the nilpotent cone are star-shaped

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

Ex.311The special linear group is totally disconnected

#totally disconnected #discrete subspace #integer matrices

Ex.312SO(2) is a circle, O(2) has two components

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

Ex.313Components of the set where the trace is non-zero

#connected components #trace #half-spaces

Ex.685Complement of a subspace

#hyperplane #codimension #connectedness #linear subspace

Ex.686The complement of O(n) is path connected

#orthogonal group #complement #singular values #path connected

Ex.687Hurwitz (stable) matrices form a connected set

#Hurwitz matrices #stability #eigenvalues #linear deformation

Ex.688Level sets of the determinant

#determinant #level sets #SL_n #scaling

Ex.689The general linear group is path connected

#polar decomposition #general linear group #path connected

Ex.690Matrices with real positive spectrum

#eigenvalues #linear deformation #path connected

Ex.691O(n) has exactly two components

#orthogonal group #components #reflections #homeomorphism

Ex.692Spectral radius balls and star-shaped sets

#spectral radius #star-shaped #norm ball

Ex.693Nilpotent matrices and idempotents of fixed rank

#nilpotent matrices #idempotents #conjugacy orbits #projections

Ex.1436The complement of the special linear group

#determinant #level sets #components #SL_n

Ex.1437Matrices of fixed rank

#rank #singular value decomposition #path connected

Ex.1438SO(n) is connected by Givens rotations

#Givens rotations #induction #special orthogonal group

Ex.1439Components of the sets of idempotents and involutions

#idempotent #involution #trace #rank #conjugacy orbits

Ex.1440Matrices of rank at least k are path connected

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

Ex.1441The special linear group is path connected (transvections)

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

Ex.1442The special linear group is path connected (polar decomposition)

#special linear group #polar decomposition #path connected

Ex.1443SO(n) is path connected (normal form)

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

Ex.2056Complex structures on the set A² = -I

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

Ex.2057Connectedness of conjugacy orbits

#conjugacy orbits #centraliser #group action #rotation matrix

Ex.2058SO(n) is a deformation retract of the general linear group

#deformation retraction #homotopy equivalence #polar decomposition

Ex.2059Matrices with n distinct real eigenvalues

#diagonalisable matrices #Weyl chamber #conjugation #path connected

Ex.2060Components of the invertible symmetric matrices

#symmetric matrices #signature #eigenvalues #Weyl inequality

Ex.2061GL(2,ℝ) is not simply connected

#fundamental group #deformation retraction #circle #homotopy

Ex.20622×2 matrices without real eigenvalues

#2x2 matrices #discriminant #complex eigenvalues #components