Norms and metric structure

34 problems · Frobenius and operator norms, equivalence of norms, balls and distances in Mₙ(ℝ).

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Ex.294The Frobenius norm comes from an inner product

#Frobenius norm #inner product #trace

Ex.295Two norms of the identity matrix

#Frobenius norm #operator norm #identity matrix

Norms and metric structure

Ex.296Entries versus Frobenius norm

#Frobenius norm #max norm #equivalence of norms

Norms and metric structure

Ex.297Every norm on M_n(R) is complete

#completeness #finite dimension #equivalence of norms

Norms and metric structure

Ex.298Column-sum and row-sum norms of a 2×2 matrix

#matrix norms #column sum #row sum #computation

Norms and metric structure

Ex.663The Frobenius norm is submultiplicative

#Frobenius norm #Cauchy-Schwarz #submultiplicativity

Norms and metric structure

Ex.664Operator norm: submultiplicativity and comparison with Frobenius

#operator norm #submultiplicativity #Frobenius norm

Norms and metric structure

Ex.665Operator norms of three basic matrices

#operator norm #nilpotent #diagonal #rotation

Norms and metric structure

Ex.666The max-entry norm is not submultiplicative

#max norm #submultiplicativity #counterexample

Norms and metric structure

Ex.667Orthogonal invariance of the Frobenius and operator norms

#orthogonal matrices #isometry #trace cyclicity

Norms and metric structure

Ex.668Operator norm of a symmetric matrix

#symmetric matrices #spectral theorem #operator norm

Norms and metric structure

Ex.669The diameter of the orthogonal group

#orthogonal group #diameter #triangle inequality

Norms and metric structure

Ex.1397The operator norm via the largest eigenvalue of Aáµ€A

#spectral norm #singular values #Rayleigh quotient

Norms and metric structure

Ex.1398The operator norm of a shear matrix

#operator norm #golden ratio #singular values

Norms and metric structure

Ex.1399The operator norm is invariant under transposition

#transpose #operator norm #duality

Norms and metric structure

Ex.1400Eigenvalues are bounded by the operator norm

#spectral radius #complex eigenvalue #operator norm

Norms and metric structure

Ex.1401The Neumann series

#Neumann series #invertibility #geometric series #Banach algebra

Norms and metric structure

Ex.1402Invertibility is stable under small perturbations

#invertible matrices #perturbation #Neumann series #open set

Norms and metric structure

Ex.1403The trace is bounded by the Frobenius norm

#trace #Cauchy-Schwarz #Frobenius norm

Norms and metric structure

Ex.1404The Schur test

#Schur test #spectral radius #induced norms

Norms and metric structure

Ex.1405Strict convexity of the Frobenius ball but not of the operator ball

#strict convexity #unit ball #parallelogram law #operator norm

Norms and metric structure

Ex.1406The rank metric

#rank #discrete topology #metric #continuity

Norms and metric structure

Ex.1407Linear functionals on M_n(R) are traces against a matrix

#linear functionals #Riesz representation #continuity

Norms and metric structure

Ex.1408Induced operator norms make M_n(R) a Banach algebra

#Banach algebra #induced norm #equivalent norms

Norms and metric structure

Ex.1409All norms on M_n(R) are equivalent

#equivalence of norms #compactness #finite dimension

Norms and metric structure

Ex.1410Distance from an invertible matrix to the singular matrices

#distance to singular matrices #singular values #Eckarte-Young #SVD

Norms and metric structure

Ex.1411The inverse blows up near singular matrices

#inverse #blow-up #distance to singular matrices

Norms and metric structure

Ex.1412The best constants comparing operator and Frobenius norms

#equivalence constants #singular values #rank one #orthogonal

Norms and metric structure

Ex.2035Hadamard's inequality for determinants

#Hadamard inequality #QR decomposition #Gram-Schmidt #determinant

Norms and metric structure

Ex.2036An upper bound of the determinant

#Hadamard inequality #AM-GM #equality case #determinant

Norms and metric structure

Ex.2037The determinant is Lipschitz on bounded sets

#determinant #Lipschitz #telescoping #Hadamard inequality

Norms and metric structure

Ex.2038The extreme points of the operator-norm unit ball

#extreme points #orthogonal group #singular value decomposition #convexity

Norms and metric structure

Ex.2039The operator-norm ball is the convex hull of O(n)

#convex hull #orthogonal group #SVD #Krein-Milman

Norms and metric structure

Ex.2040A majoration of the spectral radius

#spectral radius #Gelfand formula #symmetric matrices #powers

Norms and metric structure