Continuity of matrix maps

31 problems · Continuity of determinant, trace, powers, inverse, adjugate and other matrix maps.

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Ex.303Polynomial maps between matrix spaces are continuous

#polynomial maps #continuity #determinant #adjugate

Ex.304Matrix multiplication is jointly continuous

#multiplication #bilinear map #continuity #submultiplicativity

Continuity of matrix maps

Ex.305Inversion is a homeomorphism of GL_n

#inverse #Cramer's rule #homeomorphism #adjugate

Continuity of matrix maps

Ex.306Which natural functions of a matrix are continuous?

#continuity #rank #kernel #eigenvalues #counterexamples

Continuity of matrix maps

Ex.307GL_n is a topological group

#topological group #determinant #homomorphism #SL_n

Continuity of matrix maps

Ex.677Conjugation is continuous

#conjugation #composition of continuous maps #similarity

Continuity of matrix maps

Ex.678Lipschitz estimates for the norm and for inversion

#Lipschitz #inversion #resolvent identity #operator norm

Continuity of matrix maps

Ex.679Inversion is not uniformly continuous

#uniform continuity #inversion #singular matrices #counterexample

Continuity of matrix maps

Ex.680Matrices of negative determinant are not squares

#squares #determinant sign #surjectivity

Continuity of matrix maps

Ex.681The nearest symmetric matrix is the symmetric part

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

Continuity of matrix maps

Ex.682The characteristic polynomial map and its section

#characteristic polynomial #companion matrix #continuous section

Continuity of matrix maps

Ex.683The nilpotent cone is closed but not compact

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

Continuity of matrix maps

Ex.684A path from positive to negative determinant hits a singular matrix

#intermediate value theorem #determinant #paths #connectedness

Continuity of matrix maps

Ex.1428The derivative of the determinant (Jacobi's formula)

#derivative of determinant #Jacobi formula #adjugate #density

Continuity of matrix maps

Ex.1429The derivative of matrix inversion

#derivative #inversion #Neumann series #Fréchet derivative

Continuity of matrix maps

Ex.1430Roots of a polynomial depend continuously on its coefficients

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

Continuity of matrix maps

Ex.1431The spectral radius is continuous

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

Continuity of matrix maps

Ex.1432Rank is lower but not upper semicontinuous

#rank #semicontinuity #minors #open sets

Continuity of matrix maps

Ex.1433The rank of the adjugate matrix

#adjugate #rank #minors #kernel

Continuity of matrix maps

Ex.1434The sign of the determinant under squaring

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

Continuity of matrix maps

Ex.1435The determinant is an open map

#determinant #open map #SVD #singular matrices

Continuity of matrix maps

Ex.2046Weyl's inequality for singular values

#singular values #Courant-Fischer #perturbation #Lipschitz

Continuity of matrix maps

Ex.2047Weyl's inequality for eigenvalues of symmetric matrices

#eigenvalues #symmetric matrices #Courant-Fischer #perturbation

Continuity of matrix maps

Ex.2048The PSD square root is continuous

#matrix square root #compactness #uniqueness #PSD cone

Continuity of matrix maps

Ex.2049Polar decomposition is a homeomorphism

#polar decomposition #homeomorphism #orthogonal group #positive definite

Continuity of matrix maps

Ex.2050GL_n is homeomorphic to O(n) × upper triangular

#QR decomposition #Gram-Schmidt #Iwasawa #homeomorphism

Continuity of matrix maps

Ex.2051Cholesky: a homeomorphism from T_+ onto PD

#Cholesky decomposition #positive definite matrices #QR #homeomorphism

Continuity of matrix maps

Ex.2052The orthogonal matrix nearest to a given matrix

#orthogonal Procrustes #SVD #Frobenius norm #polar decomposition

Continuity of matrix maps

Ex.2053The Cayley transform of a skew-symmetric matrix

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

Continuity of matrix maps

Ex.2054No continuous choice of a unit eigenvector

#eigenvectors #Möbius band #connectedness #no continuous section

Continuity of matrix maps

Ex.2055SO(2) has no continuous square root

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

Continuity of matrix maps