Maths WorkshopMaths problems with full solutions

Density and closures

27 problems · Dense subsets of Mₙ(ℝ), diagonalisable matrices, closures and applications of density.

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Ex.289The closed set det(A - I) = 0 has empty interior

#hypersurface #determinant #closed set

Density and closures

Ex.648A discrete cyclic subgroup of the general linear group

#discrete subgroup #closed set #powers of a matrix

Density and closures

Ex.649Closure of the PSD matrices of rank k

#positive semidefinite #rank #closure

Density and closures

Ex.650Invertible rational matrices are dense in the general linear group

#rational matrices #density #open set

Density and closures

Ex.651Real-diagonalisable matrices are not dense

#diagonalisation #non-real eigenvalues #continuity of roots

Density and closures

Ex.652Three open dense subsets of the space of matrices

#density #discriminant #principal minors #polynomials

Density and closures

Ex.1350AB and BA have the same characteristic polynomial

#characteristic polynomial #density #similarity

Density and closures

Ex.1351Adjugate identities by density

#adjugate #determinant #density

Density and closures

Ex.1352Proper algebraic subsets of M_n(R) have measure zero

#Lebesgue measure #polynomials #Fubini #induction

Density and closures

Ex.1353The Cayley-Hamilton theorem by density

#Cayley-Hamilton #density #diagonalisation

Density and closures

Ex.1354Closure of the matrices with a non-real eigenvalue

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

Density and closures

Ex.1355Finite-order rotations are dense in SO(2)

#rotations #roots of unity #density

Density and closures

Ex.1356Idempotents of a given rank form a closed set

#idempotent #projections #trace #rank

Density and closures

Ex.1357Powers of a rotation: when are they dense?

#Kronecker #pigeonhole #rotation group #dense subgroup

Density and closures

Ex.1358Matrices with an eigenvalue of modulus one

#spectrum #closed sets #empty interior #scaling

Density and closures

Ex.1359Matrices with an LU factorisation

#LU decomposition #leading principal minors #density

Density and closures

Ex.1360Nonderogatory matrices form an open dense set

#minimal polynomial #nonderogatory #density

Density and closures

Ex.1361Orthogonally invariant functions of symmetric matrices

#spectral theorem #invariant functions #eigenvalues

Density and closures

Ex.1362density in the special linear group

#special linear group #transvections #density

Density and closures

Ex.1363Rational rotation matrices are dense in SO(2)

#rational points #circle #stereographic projection #Pythagorean triples

Density and closures

Ex.1364The closure of the Hurwitz matrices

#Hurwitz #stability #spectrum #closure

Density and closures

Ex.1365The complement of a polynomial zero set is open and dense

#polynomials #density #Taylor expansion

Density and closures

Ex.1366The orbit of a Jordan block accumulates on the identity

#conjugacy orbit #closure #Jordan block

Density and closures

Ex.1367The set of real normal matrices

#normal matrices #closed sets #subspaces

Density and closures

Ex.2010Orbits of real-diagonalisable matrices are closed

#conjugacy orbit #minimal polynomial #closed sets

Density and closures

Ex.2011Orbits of non-diagonalisable matrices are not closed

#conjugacy orbit #Jordan form #scaling #closure

Density and closures

Ex.2012The regular nilpotent orbit is dense in the nilpotent cone

#nilpotent matrices #Jordan form #orbit closure

Density and closures