Topology of Mn(ℝ)

280 problems · Norms, open and closed sets, compactness, connectedness, density and the exponential in the space of real square matrices.

Search problems Level

Ex.287Odd-sized real matrices have a real eigenvalue

#intermediate value theorem #characteristic polynomial #eigenvalue

Applications and the case n = 2

Ex.288The inverse does not extend continuously across singular matrices

#inversion #unboundedness #continuity #extension

Applications and the case n = 2

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

#hypersurface #determinant #closed set

Density and closures

Ex.290Convergence of the matrix exponential and its first bounds

#matrix exponential #absolute convergence #submultiplicative norm

Exponential, logarithm and group topology

Ex.291The exponential of a sum: commuting and non-commuting matrices

#matrix exponential #commutation #counterexample

Exponential, logarithm and group topology

Ex.292Every norm sphere of M_n(R) is a sphere

#norms #sphere #homeomorphism #radial projection

Rank, spectrum and norms

Ex.293Trace and determinant bounded by the operator norm

#operator norm #singular values #trace #determinant

Rank, spectrum and norms

Ex.294The Frobenius norm comes from an inner product

#Frobenius norm #inner product #trace

Norms and metric structure

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.299The groups GL_n and SL_n: open and closed

#open sets #closed sets #determinant #GL_n #SL_n

Open, closed, interior, boundary

Ex.300Proper subspaces are closed with empty interior

#subspaces #interior #closed sets #finite dimension

Open, closed, interior, boundary

Ex.301Closed or open? Three classical subspaces

#subspaces #symmetric matrices #trace #triangular matrices

Open, closed, interior, boundary

Ex.302Integer matrices are closed and discrete, rational ones dense

#integer matrices #discrete sets #density #rationals

Open, closed, interior, boundary

Ex.303Polynomial maps between matrix spaces are continuous

#polynomial maps #continuity #determinant #adjugate

Continuity of matrix maps

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.308Components of the linear group

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

Connectedness

Ex.309Convex sets of matrices are path connected

#connectedness #convex sets #path connected #matrices

Connectedness

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

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

Connectedness

Ex.311The special linear group is totally disconnected

#totally disconnected #discrete subspace #integer matrices

Connectedness

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

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

Connectedness

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

#connected components #trace #half-spaces

Connectedness

Ex.629Continuity of the determinant

#determinant #continuity #polynomial

Topology of Mn(ℝ)

Ex.630Convergence in $M_n(\mathbb{R})$ via entries

#matrix convergence #entrywise convergence #sequence of matrices

Topology of Mn(ℝ)

Ex.631Summary of the topology of $M_n(\mathbb{R})$

#summary #matrix topology #M_n(R)

Topology of Mn(ℝ)

High school

Ex.646A 2×2 matrix with positive entries has a positive eigenvector

#intermediate value theorem #Perron-Frobenius #eigenvector

Applications and the case n = 2

Ex.647Convexity of Î»_max and the spectral norm on symmetric matrices

#convexity #Rayleigh quotient #norm #symmetric matrices

Applications and the case n = 2

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.653The closure of a subgroup of GL_n is a subgroup

#topological group #closure #subgroup

Exponential, logarithm and group topology

Ex.654The derivative of the matrix exponential and the one-parameter group property

#matrix exponential #derivative #homomorphism

Exponential, logarithm and group topology

Ex.655The exponential is not injective: rotations as exponentials

#matrix exponential #rotation #kernel

Exponential, logarithm and group topology

Ex.656Existence of a closest matrix of rank at most r

#low rank approximation #compactness #coercivity

Rank, spectrum and norms

Ex.657Singular matrices have measure zero

#Lebesgue measure #zero set of a polynomial #Fubini #determinant

Rank, spectrum and norms

Ex.658The singular value map is continuous and onto the ordered cone

#singular values #Weyl inequality #continuity #surjectivity

Rank, spectrum and norms

Ex.659Strictly diagonally dominant matrices are invertible

#diagonal dominance #Gershgorin #open sets #invertibility

Rank, spectrum and norms

Ex.660The numerical range of a real matrix is an interval

#Rayleigh quotient #connectedness #compactness #symmetric part

Rank, spectrum and norms

Ex.661The Moore-Penrose pseudo-inverse is discontinuous

#pseudo-inverse #discontinuity #rank #singular value decomposition

Rank, spectrum and norms

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.670Open ball, closed ball and sphere in the operator norm

#operator norm #ball #closure #boundary

Open, closed, interior, boundary

Ex.671Positive definite matrices: open cone with closure PSD

#positive definite #cones #eigenvalues #interior

Open, closed, interior, boundary

Ex.672PSD matrices have empty interior in all of M_n

#positive semidefinite #symmetric matrices #interior

Open, closed, interior, boundary

Ex.673Nilpotent matrices and idempotents are closed

#nilpotent #idempotent #algebraic sets #nowhere dense

Open, closed, interior, boundary

Ex.674GL_n is dense and its boundary is the singular set

#invertible matrices #density #boundary #determinant

Open, closed, interior, boundary

Ex.675Matrices with no zero entry: open and dense

#nowhere dense #hyperplanes #open dense sets

Open, closed, interior, boundary

Ex.676Entrywise nonnegative matrices: a closed cone with interior

#cones #convexity #interior #nonnegative matrices

Open, closed, interior, boundary

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.685Complement of a subspace

#hyperplane #codimension #connectedness #linear subspace

Connectedness

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

#orthogonal group #complement #singular values #path connected

Connectedness

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

#Hurwitz matrices #stability #eigenvalues #linear deformation

Connectedness

Ex.688Level sets of the determinant

#determinant #level sets #SL_n #scaling

Connectedness

Ex.689The general linear group is path connected

#polar decomposition #general linear group #path connected

Connectedness

Ex.690Matrices with real positive spectrum

#eigenvalues #linear deformation #path connected

Connectedness

Ex.691O(n) has exactly two components

#orthogonal group #components #reflections #homeomorphism

Connectedness

Ex.692Spectral radius balls and star-shaped sets

#spectral radius #star-shaped #norm ball

Connectedness

Ex.693Nilpotent matrices and idempotents of fixed rank

#nilpotent matrices #idempotents #conjugacy orbits #projections

Connectedness

Ex.694The idempotents 0 and I are isolated points

#idempotents #projections #isolated points #trace and rank

Applications and the case n = 2

Ex.695Square-zero matrices form a path connected set

#nilpotent matrices #star-shaped sets #path connectedness #cones

Applications and the case n = 2

Ex.696The operator-norm distance between two rotations is a chord length

#M2(R) #rotations #operator norm #chord length

Applications and the case n = 2

Ex.1286A non-compact closed subspace: trace-zero matrices

#closed unbounded #trace #non-compactness

Topology of Mn(ℝ)

Ex.1287Completeness of $M_n(\mathbb{R})$

#completeness #Cauchy sequence #identification with R^{n^2}

Topology of Mn(ℝ)

Ex.1288Interior and boundary of $GL_n(\mathbb{R})$

#interior #boundary #GL_n

Topology of Mn(ℝ)

Ex.1289Equivalent metrics on $M_n(\mathbb{R})$

#equivalent metrics #norms on matrices #finite dimension

Topology of Mn(ℝ)

Ex.1290Submultiplicativity of matrix norms

#submultiplicative norms #operator norm #matrix norm

Topology of Mn(ℝ)

Ex.1291Separability of $M_n(\mathbb{R})$

#separability #dense countable subset #rational matrices

Topology of Mn(ℝ)

Ex.1341det(I + AB) = det(I + BA) by density

#density #determinant #polynomial identity

Applications and the case n = 2

Ex.1342Exponential coordinates on SO(3)

#exponential map #SO(3) #Rodrigues formula #homeomorphism

Applications and the case n = 2

Ex.1343Rank-one orthogonal projections form the projective space

#projective space #projections #compact Hausdorff #homeomorphism

Applications and the case n = 2

Ex.1344Odd-dimensional rotations have an axis

#orthogonal group #determinant #eigenvalue 1 #Euler's theorem

Applications and the case n = 2

Ex.1345The quotient of M_n(R) by conjugation is not Hausdorff

#quotient topology #conjugation #Jordan block #separation axioms

Applications and the case n = 2

Ex.1346AAáµ€ and Aáµ€A have the same characteristic polynomial

#characteristic polynomial #density #symmetric matrices

Applications and the case n = 2

Ex.1347The Grassmannian of orthogonal projections is compact and connected

#Grassmannian #compactness #connectedness #SO(n) orbit

Applications and the case n = 2

Ex.1348Sym_n/O(n) is a closed Weyl chamber

#orbit space #eigenvalues #Weyl chamber #quotient

Applications and the case n = 2

Ex.1349The rotation angle of a 3D rotation is continuous

#SO(3) #rotation angle #trace #normal form

Applications and the case n = 2

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.1368A matrix of positive determinant that is not an exponential

#matrix exponential #surjectivity #commutation #diagonal matrices

Exponential, logarithm and group topology

Ex.1369Centres of GL_n and SL_n, and discrete normal subgroups

#centre #discrete subgroup #connected group

Exponential, logarithm and group topology

Ex.1370The determinant of an exponential: det e^A = e^(tr A)

#matrix exponential #determinant #trace #triangularisation

Exponential, logarithm and group topology

Ex.1371exp is a homeomorphism from symmetric to positive definite matrices

#symmetric matrices #positive definite #homeomorphism #proper map

Exponential, logarithm and group topology

Ex.1372The quotients GL_n/O(n) and SL_n/SO(n)

#homogeneous space #quotient topology #positive definite #traceless

Exponential, logarithm and group topology

Ex.1373The polar-exponential decomposition of GL_n

#polar decomposition #homeomorphism #orthogonal group #GL_n

Exponential, logarithm and group topology

Ex.1374SO(n)/SO(n-1) is the sphere

#orbit map #quotient #sphere #compact Hausdorff

Exponential, logarithm and group topology

Ex.1375The exponential maps skew-symmetric matrices onto SO(n)

#orthogonal group #skew-symmetric #normal form #connectedness

Exponential, logarithm and group topology

Ex.1376The logarithm near the identity and local inverse of exp

#matrix logarithm #local homeomorphism #power series

Exponential, logarithm and group topology

Ex.1377Connected components of the projective linear group

#projective group #quotient #determinant #parity

Exponential, logarithm and group topology

Ex.1378Nilpotent matrices and unipotent matrices

#nilpotent #unipotent #polynomial inverse #exponential

Exponential, logarithm and group topology

Ex.1379Topology of the groups of upper triangular matrices

#triangular matrices #connected components #contractible #unipotent

Exponential, logarithm and group topology

Ex.1380Eigenvalues are only Hölder continuous

#eigenvalues #Hölder continuity #Jordan block #perturbation

Rank, spectrum and norms

Ex.1381Semicontinuity of the inertia of symmetric matrices

#inertia #symmetric matrices #Weyl #semicontinuity

Rank, spectrum and norms

Ex.1382Inversion reverses the order of positive definite matrices

#Loewner order #positive definite #inverse #square root

Rank, spectrum and norms

Ex.1383Matrices with positive definite symmetric part

#positive definite #convex cone #eigenvalues #real part

Rank, spectrum and norms

Ex.1384Power-bounded matrices do not form a closed set

#power bounded #Jordan block #closed sets

Rank, spectrum and norms

Ex.1385Power sums determine the characteristic polynomial

#Newton identities #power sums #characteristic polynomial #continuity

Rank, spectrum and norms

Ex.1386The cone of rank-one matrices

#rank one #closed cone #connectedness #minors

Rank, spectrum and norms

Ex.1387Semicontinuity of the minimal polynomial degree and of the centraliser

#lower semicontinuity #minimal polynomial #centraliser #rank

Rank, spectrum and norms

Ex.1388Where is {det = 0} a manifold?

#determinant #implicit function theorem #singular point #cone #local homology

Rank, spectrum and norms

Ex.1389The discriminant of a 2×2 matrix and a cone

#discriminant #2x2 matrices #convex cone #complex eigenvalues

Rank, spectrum and norms

Ex.1390The set of matrices whose powers converge is neither open nor closed

#powers #convergence #open sets #closed sets

Rank, spectrum and norms

Ex.1391The spectral mapping theorem for polynomials

#spectrum #triangularisation #polynomial of a matrix

Rank, spectrum and norms

Ex.1392A^k converges to 0 if and only if the spectral radius is less than 1

#spectral radius #Jordan form #powers of a matrix

Rank, spectrum and norms

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.1413The closure of {det > 0}

#determinant #singular value decomposition #closure #boundary

Open, closed, interior, boundary

Ex.1414Lower-rank matrices form a closed set, higher-rank an open one

#rank #minors #semicontinuity #nowhere dense

Open, closed, interior, boundary

Ex.1415Matrices of rank exactly r: neither open nor closed

#rank #closure #SVD #semicontinuity

Open, closed, interior, boundary

Ex.1416Matrices with a real eigenvalue: closed set, interior for n = 2

#real eigenvalues #closed sets #discriminant #2x2 matrices

Open, closed, interior, boundary

Ex.1417Real-diagonalisable matrices: neither open nor closed

#diagonalisable #Jordan blocks #counterexamples

Open, closed, interior, boundary

Ex.1418Matrices with n distinct eigenvalues form an open set

#discriminant #distinct eigenvalues #open sets #characteristic polynomial

Open, closed, interior, boundary

Ex.1419The orthogonal group is closed and nowhere dense

#orthogonal group #spheres #nowhere dense #Frobenius norm

Open, closed, interior, boundary

Ex.1420Involutions form a closed unbounded set

#involutions #closed sets #unbounded #counterexample

Open, closed, interior, boundary

Ex.1421The boundary of the positive definite cone

#positive definite #boundary #symmetric matrices #determinant

Open, closed, interior, boundary

Ex.1422Orthogonal projections: compact with empty interior in Sym_n

#projections #compactness #symmetric matrices #nowhere dense

Open, closed, interior, boundary

Ex.1423An open subgroup of GL_n is also closed

#topological groups #open subgroup #cosets #GL_n

Open, closed, interior, boundary

Ex.1424Interior and boundary of {det > 1}

#determinant #interior #boundary #level sets

Open, closed, interior, boundary

Ex.1425The commutant of a non-scalar matrix is a thin subspace

#commutant #subspaces #scalar matrices #nowhere dense

Open, closed, interior, boundary

Ex.1426Matrices with rational trace: dense with empty interior

#trace #density #rationals #open maps

Open, closed, interior, boundary

Ex.1427Countably many proper subspaces never cover M_n

#Baire category theorem #subspaces #covering

Open, closed, interior, boundary

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.1436The complement of the special linear group

#determinant #level sets #components #SL_n

Connectedness

Ex.1437Matrices of fixed rank

#rank #singular value decomposition #path connected

Connectedness

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

#Givens rotations #induction #special orthogonal group

Connectedness

Ex.1439Components of the sets of idempotents and involutions

#idempotent #involution #trace #rank #conjugacy orbits

Connectedness

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

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

Connectedness

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

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

Connectedness

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

#special linear group #polar decomposition #path connected

Connectedness

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

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

Connectedness

Ex.1445Complex coordinates (z, w) on M_2(R): determinant and singular values

#M_2(R) #complex coordinates #singular values #determinant #Frobenius norm

Applications and the case n = 2

Ex.1446Conjugation by rotations acts on w by e^{2iÎ²}

#M_2 #conjugation #rotations #complex coordinates #invariants

Applications and the case n = 2

Ex.1447Matrices of finite order are dense in SO(n)

#SO(n) #finite order #normal form of orthogonal matrices #density

Applications and the case n = 2

Ex.1448Real eigenvalues of a matrix

#M_2 #eigenvalues #discriminant #complex coordinates #real spectrum

Applications and the case n = 2

Ex.1449Rotations of finite order are dense in SO(3)

#rotations #SO(3) #density #Euler's theorem #finite order

Applications and the case n = 2

Ex.1450Signature is constant on a rank stratum of symmetric matrices

#signature #Sylvester's law of inertia #connected components #symmetric matrices

Applications and the case n = 2

Ex.1451The distance to the singular matrices in M_2(R)

#M_2 #distance to singular matrices #smallest singular value #operator norm

Applications and the case n = 2

Ex.1452The singular matrices of M_2(R) form a cone over a torus

#M_2 #singular matrices #rank one #torus #cone

Applications and the case n = 2

Ex.1453The shapes of GL(2,R), SL(2,R) and {det < 0}

#M_2 #polar coordinates #homeomorphism #circle #GL_2 #SL_2

Applications and the case n = 2

Ex.1454The operator-norm ball and O(2) in complex coordinates

#M_2 #operator norm #extreme points #O(2) #unit ball

Applications and the case n = 2

Ex.1455Schur-stable 2×2 matrices: the Jury conditions and connectedness

#spectral radius #Schur stability #Jury criterion #M2(R) #path connectedness

Applications and the case n = 2

Ex.1456The cone of positive definite 2×2 matrices is a light cone

#positive definite matrices #convex cone #Sym2 #boundary

Applications and the case n = 2

Ex.1457The convex hull of SO(2) is a flat disc

#M2(R) #convex hull #rotations #complex coordinates

Applications and the case n = 2

Ex.1458The 2×2 Hurwitz matrices form a path-connected set

#M2(R) #Hurwitz stability #path connectedness #trace and determinant

Applications and the case n = 2

Ex.1459Rank-one idempotents and involutions of M_2(R) are cylinders

#M_2(R) #complex coordinates #idempotents #involutions #trace and determinant

Applications and the case n = 2

Ex.1460The distance from a 2×2 matrix to the rotation group

#nearest rotation #Frobenius norm #SO(2) #M2(R) #Procrustes

Applications and the case n = 2

Ex.1461The nonzero nilpotents of M_2(R) form two components

#M_2(R) #nilpotent matrices #connected components #complex coordinates

Applications and the case n = 2

Ex.1462The square roots of A in M_2(R) form two disjoint planes

#M_2(R) #complex coordinates #complex structures #connected components

Applications and the case n = 2

Ex.1963Classical closed subgroups of $GL_n(\mathbb{R})$

#closed subgroups #GL_n #SL_n #O_n #determinant

Topology of Mn(ℝ)

Ex.1964Compactness of orthogonal and orthogonal groups

#compactness #O_n #SO_n #boundedness

Topology of Mn(ℝ)

Ex.1965Continuity of eigenvalues (statement)

#eigenvalues #continuity #roots of polynomial

Topology of Mn(ℝ)

Ex.1966A homeomorphism of $M_n(\mathbb{R})$

#homeomorphism #matrix map #inverse map

Topology of Mn(ℝ)

Ex.1967Continuity of the matrix exponential

#matrix exponential #continuity #series

Topology of Mn(ℝ)

Ex.1968$GL_n(\mathbb{R})$ is open in $M_n(\mathbb{R})$

#GL_n #open set #determinant continuity

Topology of Mn(ℝ)

Ex.2000A²+B² is singular when A³ = B³ and A²B = B²A

#kernel #algebraic identity #Putnam

Applications and the case n = 2

Ex.2001The determinant of a block matrix with commuting lower blocks

#density #block matrices #Schur complement

Applications and the case n = 2

Ex.2002Invertible skew-symmetric matrices have two components

#Pfaffian #connected components #skew-symmetric matrices

Applications and the case n = 2

Ex.2003Rotations by Ï€ form a projective plane; the Cayley transform covers the rest

#SO(3) #Cayley transform #projective plane #cut locus

Applications and the case n = 2

Ex.2004adj(adj A) = (det A)^(n-2) A

#adjugate #density #polynomial identity

Applications and the case n = 2

Ex.2005det(A² + B²) is positive for commuting real matrices

#complexification #determinant #commuting matrices

Applications and the case n = 2

Ex.2006Ky Fan's maximum principle

#Ky Fan #eigenvalues #orthogonal projections #convexity

Applications and the case n = 2

Ex.2007M_n(R)/(O(n) . O(n)) is the space of singular values

#singular values #SVD #orbit space #Weyl inequality

Applications and the case n = 2

Ex.2008M_n(R)/O(n) is the cone of positive semidefinite matrices

#quotient space #proper maps #positive semidefinite #orbit space

Applications and the case n = 2

Ex.2009Symmetric matrices with distinct eigenvalues form a connected set

#connectedness #symmetric matrices #spectral theorem #SO(n)

Applications and the case n = 2

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

Ex.2013A subgroup of SO(4) that is not closed

#irrational winding #torus #Kronecker #non-closed subgroup

Exponential, logarithm and group topology

Ex.2014Compact subgroups of SL_n are conjugate into SO(n)

#compact group #Haar measure #invariant inner product #conjugation

Exponential, logarithm and group topology

Ex.2015Every continuous one-parameter subgroup is e^(tA)

#one-parameter subgroup #regularity #integration trick

Exponential, logarithm and group topology

Ex.2016SO(3) is homeomorphic to the 3-sphere

#SO(3) #quaternions #real projective space #double cover

Exponential, logarithm and group topology

Ex.2017A neighbourhood of the identity generates the connected group

#generating set #open subgroup #connected group #GL_n^+

Exponential, logarithm and group topology

Ex.2018The fundamental group of SO(3)

#fundamental group #covering space #SO(3) #deck transformations

Exponential, logarithm and group topology

Ex.2019The Lie algebra of a closed subgroup is a linear subspace

#Lie algebra #closed subgroup #Lie product formula

Exponential, logarithm and group topology

Ex.2020The Lie product formula

#Lie product formula #Trotter #norm estimates #telescoping

Exponential, logarithm and group topology

Ex.2021Which matrices of the special linear group are exponentials?

#SL_2 #trace #Cayley-Hamilton #exponential image

Exponential, logarithm and group topology

Ex.2022A traceless matrix has a zero diagonal in some orthonormal basis

#trace #numerical range #orthonormal basis #induction

Rank, spectrum and norms

Ex.2023The fibres of (trace, determinant) on 2×2 matrices

#conjugacy classes #2x2 matrices #hyperboloid #trace #determinant

Rank, spectrum and norms

Ex.2024Dobrushin's contraction for positive stochastic matrices

#stochastic matrices #Dobrushin #oscillation #convergence of powers

Rank, spectrum and norms

Ex.2025Eigenvalues of normal matrices move by at most the size of the perturbation

#normal matrices #Bauer-Fike #Weyl #unitary diagonalisation

Rank, spectrum and norms

Ex.2026A stochastic matrix has a stationary distribution

#stochastic matrices #compactness #Cesàro averages #Markov chains

Rank, spectrum and norms

Ex.2027Gelfand's spectral radius formula

#spectral radius #Gelfand formula #norms #powers

Rank, spectrum and norms

Ex.2028Lyapunov's criterion for Hurwitz matrices

#Hurwitz #Lyapunov equation #stability #positive definite

Rank, spectrum and norms

Ex.2029The spectral radius is monotone on nonnegative matrices

#spectral radius #nonnegative matrices #Gelfand formula #monotonicity

Rank, spectrum and norms

Ex.2030Sylvester's criterion and the openness of the positive cone

#Sylvester criterion #positive definite #LDL decomposition #leading minors

Rank, spectrum and norms

Ex.2031The Bauer-Fike theorem

#Bauer-Fike #perturbation of eigenvalues #condition number #diagonalisable

Rank, spectrum and norms

Ex.2032The Eckart-Young theorem in the operator norm

#Eckart-Young #low rank approximation #singular values #dimension counting

Rank, spectrum and norms

Ex.2033Matrices with two distinct real eigenvalues retract onto a circle

#homotopy equivalence #deformation retraction #2x2 matrices #discriminant

Rank, spectrum and norms

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

Ex.2041The set {Ï(A) < 1}: open, with boundary {Ï = 1}

#spectral radius #continuity of eigenvalues #closure #boundary

Open, closed, interior, boundary

Ex.2042The interior of the real-diagonalisable matrices

#diagonalisable #interior #distinct eigenvalues #Jordan blocks

Open, closed, interior, boundary

Ex.2043The closure of the real-diagonalisable matrices

#diagonalisable #closure #triangularisation #continuity of roots

Open, closed, interior, boundary

Ex.2044Matrices with a repeated eigenvalue are rare

#discriminant #algebraic sets #polynomials #nowhere dense

Open, closed, interior, boundary

Ex.2045Symmetric matrices with distinct eigenvalues are open and dense

#symmetric matrices #spectral theorem #genericity #distinct eigenvalues

Open, closed, interior, boundary

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

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

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

Connectedness

Ex.2057Connectedness of conjugacy orbits

#conjugacy orbits #centraliser #group action #rotation matrix

Connectedness

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

#deformation retraction #homotopy equivalence #polar decomposition

Connectedness

Ex.2059Matrices with n distinct real eigenvalues

#diagonalisable matrices #Weyl chamber #conjugation #path connected

Connectedness

Ex.2060Components of the invertible symmetric matrices

#symmetric matrices #signature #eigenvalues #Weyl inequality

Connectedness

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

#fundamental group #deformation retraction #circle #homotopy

Connectedness

Ex.20622×2 matrices without real eigenvalues

#2x2 matrices #discriminant #complex eigenvalues #components

Connectedness

Ex.2063Ordered real eigenvalues are continuous on the real-rooted matrices

#continuity of roots #ordered eigenvalues #Rouché #real-rooted matrices

Applications and the case n = 2

Ex.2064The complex structures A² =-I on form exactly two components

#conjugacy orbits #complex structures #orientation #connected components #GL_n

Applications and the case n = 2

Ex.2065The operator-norm unit sphere of M_2(R) is a 3-sphere

#operator norm #unit sphere #M_2(R) #radial projection #complex coordinates

Applications and the case n = 2

Ex.2066The orbit of a regular nilpotent matrix is connected iff n is odd

#nilpotent matrices #conjugacy orbits #cyclic vectors #connected components

Applications and the case n = 2

Ex.2234Distance to $O_n$ and polar decomposition (statement)

#distance to O_n #singular values #polar decomposition

Topology of Mn(ℝ)

Ex.2241Continuous homomorphisms from GL(n,R) to R

#homomorphism #commutator subgroup #determinant #continuous characters

Applications and the case n = 2

Ex.2242No continuous square root on GL(2,R)

#square root #rotations #commutant #non-existence

Applications and the case n = 2

Ex.2243The Dirac belt trick: a 2Ï€ rotation is non-trivial, a 4Ï€ rotation is trivial

#homotopy #lifting #SO(3) #quaternions #Dirac belt

Exponential, logarithm and group topology

Ex.2244The square root is operator monotone

#Loewner order #operator monotone #square root #positive semidefinite

Rank, spectrum and norms