Applications and the case n = 2

50 problems · Density arguments, closed-form results and detailed study of 2×2 matrices.

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

#intermediate value theorem #characteristic polynomial #eigenvalue

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

#inversion #unboundedness #continuity #extension

Applications and the case n = 2

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.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.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.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.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.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.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