Maths WorkshopMaths problems with full solutions

Open, closed, interior, boundary

31 problems · Which sets of matrices are open or closed, and their interiors and boundaries.

Ex.299The groups GL_n and SL_n: open and closed
#open sets #closed sets #determinant #GL_n #SL_n
Open, closed, interior, boundary
L2
L2
Ex.300Proper subspaces are closed with empty interior
#subspaces #interior #closed sets #finite dimension
Open, closed, interior, boundary
L2
L2
Ex.301Closed or open? Three classical subspaces
#subspaces #symmetric matrices #trace #triangular matrices
Open, closed, interior, boundary
L2
L2
Ex.302Integer matrices are closed and discrete, rational ones dense
#integer matrices #discrete sets #density #rationals
Open, closed, interior, boundary
L2
L2
Ex.670Open ball, closed ball and sphere in the operator norm
#operator norm #ball #closure #boundary
Open, closed, interior, boundary
L2
L2
Ex.671Positive definite matrices: open cone with closure PSD
#positive definite #cones #eigenvalues #interior
Open, closed, interior, boundary
L2
L2
Ex.672PSD matrices have empty interior in all of M_n
#positive semidefinite #symmetric matrices #interior
Open, closed, interior, boundary
L2
L2
Ex.673Nilpotent matrices and idempotents are closed
#nilpotent #idempotent #algebraic sets #nowhere dense
Open, closed, interior, boundary
L2
L2
Ex.674GL_n is dense and its boundary is the singular set
#invertible matrices #density #boundary #determinant
Open, closed, interior, boundary
L2
L2
Ex.675Matrices with no zero entry: open and dense
#nowhere dense #hyperplanes #open dense sets
Open, closed, interior, boundary
L2
L2
Ex.676Entrywise nonnegative matrices: a closed cone with interior
#cones #convexity #interior #nonnegative matrices
Open, closed, interior, boundary
L2
L2
Ex.1413The closure of {det > 0}
#determinant #singular value decomposition #closure #boundary
Open, closed, interior, boundary
L2
L2
Ex.1414Lower-rank matrices form a closed set, higher-rank an open one
#rank #minors #semicontinuity #nowhere dense
Open, closed, interior, boundary
L3
L3
Ex.1415Matrices of rank exactly r: neither open nor closed
#rank #closure #SVD #semicontinuity
Open, closed, interior, boundary
L3
L3
Ex.1416Matrices with a real eigenvalue: closed set, interior for n = 2
#real eigenvalues #closed sets #discriminant #2x2 matrices
Open, closed, interior, boundary
L3
L3
Ex.1417Real-diagonalisable matrices: neither open nor closed
#diagonalisable #Jordan blocks #counterexamples
Open, closed, interior, boundary
L3
L3
Ex.1418Matrices with n distinct eigenvalues form an open set
#discriminant #distinct eigenvalues #open sets #characteristic polynomial
Open, closed, interior, boundary
L3
L3
Ex.1419The orthogonal group is closed and nowhere dense
#orthogonal group #spheres #nowhere dense #Frobenius norm
Open, closed, interior, boundary
L3
L3
Ex.1420Involutions form a closed unbounded set
#involutions #closed sets #unbounded #counterexample
Open, closed, interior, boundary
L3
L3
Ex.1421The boundary of the positive definite cone
#positive definite #boundary #symmetric matrices #determinant
Open, closed, interior, boundary
L3
L3
Ex.1422Orthogonal projections: compact with empty interior in Sym_n
#projections #compactness #symmetric matrices #nowhere dense
Open, closed, interior, boundary
L3
L3
Ex.1423An open subgroup of GL_n is also closed
#topological groups #open subgroup #cosets #GL_n
Open, closed, interior, boundary
L3
L3
Ex.1424Interior and boundary of {det > 1}
#determinant #interior #boundary #level sets
Open, closed, interior, boundary
L3
L3
Ex.1425The commutant of a non-scalar matrix is a thin subspace
#commutant #subspaces #scalar matrices #nowhere dense
Open, closed, interior, boundary
L3
L3
Ex.1426Matrices with rational trace: dense with empty interior
#trace #density #rationals #open maps
Open, closed, interior, boundary
L3
L3
Ex.1427Countably many proper subspaces never cover M_n
#Baire category theorem #subspaces #covering
Open, closed, interior, boundary
L3
L3
Ex.2041The set {ρ(A) < 1}: open, with boundary {ρ = 1}
#spectral radius #continuity of eigenvalues #closure #boundary
Open, closed, interior, boundary
M1
M1
Ex.2042The interior of the real-diagonalisable matrices
#diagonalisable #interior #distinct eigenvalues #Jordan blocks
Open, closed, interior, boundary
M1
M1
Ex.2043The closure of the real-diagonalisable matrices
#diagonalisable #closure #triangularisation #continuity of roots
Open, closed, interior, boundary
M1
M1
Ex.2044Matrices with a repeated eigenvalue are rare
#discriminant #algebraic sets #polynomials #nowhere dense
Open, closed, interior, boundary
M1
M1
Ex.2045Symmetric matrices with distinct eigenvalues are open and dense
#symmetric matrices #spectral theorem #genericity #distinct eigenvalues
Open, closed, interior, boundary
M1
M1