Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

All ODE PDE abelian analyticity anova baire calculus of variations canonical forms cauchy-riemann compactness completeness complex conformal connectedness continuity contour integration convergence differentiation distributions fields fixed point galois green groups harmonic injectivity integral equations integration laurent limsup linear algebra liouville lp markov measure mechanics metric spaces multivariable multivariate numerical p-groups permutations polynomials power series probability quadratic forms real analysis regression residues rings sampling separability separation sequences series singularities stability statistics sylow testing topology uniform convergence wronskian zeros

#1 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic polynomial ⇒ similar” — false

Counterexample: 0 matrix and [[0,1],[0,0]]

Both have char poly $x^{2}$ , different minimal polynomials (x vs $x^{2})$ .

linear algebracanonical forms

#2 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Diagonalisable ⇒ invertible” — false

Counterexample locked — unlock with Notes + PYQ

linear algebra

#3 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Real matrix with real eigenvalues is diagonalisable” — false

Counterexample: [[1,1],[0,1]]

Eigenvalue 1 with geometric multiplicity 1 < algebraic 2.

linear algebra

#4 · Analysis & Linear Algebra › Determinants and Matrix Tricks

“AB = I ⇒ BA = I for all matrices” — false

Counterexample locked — unlock with Notes + PYQ

linear algebra

#5 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“Every real symmetric matrix is positive definite if det > 0” — false

Counterexample locked — unlock with Notes + PYQ

linear algebraquadratic forms

#6 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic and minimal polynomial ⇒ similar” — false

Counterexample: 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

Both have $χ = x^{4}$ and $m = x^{2}$ , but different numbers of blocks (ranks 2 vs 1).

linear algebracanonical forms

#7 · Analysis & Linear Algebra › Vector Spaces and Linear Maps

“AB and BA have the same minimal polynomial” — false

Counterexample: A = [[0,1],[0,0]], B = [[0,0],[0,1]]

AB = A has minimal polynomial $x^{2}$ , BA = 0 has x. The characteristic polynomials do agree.

linear algebra

#8 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Commuting matrices are simultaneously diagonalisable” — false

Counterexample: A = B = [[0,1],[0,0]]

They commute but neither is diagonalisable. Need each to be diagonalisable first.

linear algebra

#9 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Real matrix diagonalisable over ℂ ⇒ diagonalisable over ℝ” — false

Counterexample locked — unlock with Notes + PYQ

linear algebra

#10 · Analysis & Linear Algebra › Vector Spaces and Linear Maps

“An injective linear operator on a vector space is surjective” — false

Counterexample: The right shift on ℓ²: (x₁, x₂, …) ↦ (0, x₁, x₂, …)

Injective but misses everything with a non-zero first coordinate. Rank–nullity needs finite dimension.

linear algebra

#11 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Every real matrix has a Jordan form over ℝ” — false

Counterexample: The rotation [[0,−1],[1,0]]

Its eigenvalues ±i are not real; over $R$ one uses the real Jordan form with a 2×2 rotation block.

linear algebra

#12 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Two matrices with the same characteristic and minimal polynomials are similar” — false

Counterexample locked — unlock with Notes + PYQ

linear algebra

#13 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“A real matrix with all real eigenvalues is orthogonally diagonalisable” — false

Counterexample: [[1,1],[0,1]]

Eigenvalue 1 twice but only one eigenvector; orthogonal diagonalisability requires symmetry.

linear algebra

#14 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“det A > 0 implies A is positive definite” — false

Counterexample: A = diag(−1, −1)

det = 1 > 0 but both eigenvalues are negative. All leading principal minors must be positive.

linear algebraquadratic forms

#15 · Analysis & Linear Algebra › Determinants and Matrix Tricks

“There exist matrices with AB − BA = I” — false

Counterexample: Impossible over ℝ or ℂ in finite dimensions

trace(AB − BA) = 0 but trace(I) = n ≠ 0. (It is possible for unbounded operators — the Heisenberg relation.)

linear algebra