GATE MA cheat sheet

Linear algebra: eigenvalues of A satisfy det(A − λI) = 0; trace(A) = Σλᵢ and det(A) = Πλᵢ, so a matrix is singular iff some eigenvalue is 0.

A real symmetric matrix has real eigenvalues and an orthogonal set of eigenvectors; a matrix is diagonalisable iff it has a full set of linearly independent eigenvectors.

Rank–nullity: rank(A) + nullity(A) = number of columns of A; rank(A) equals the number of nonzero singular values of A.

Cauchy–Riemann: f = u + iv analytic iff uₓ = v_y and u_y = −vₓ (plus continuity of partials); harmonic u satisfies u_xx + u_yy = 0.

Residue theorem: ∮C f(z)dz = 2πi Σ Res(f, z_k) for poles inside C; a simple pole has Res = lim{z→z₀}(z − z₀)f(z).

Laplace transform facts: L{f′} = sF(s) − f(0), L{e^{at}} = 1/(s − a), L{sin at} = a/(s² + a²), L{t^n} = n!/s^{n+1}.

First-order ODE y′ + P(x)y = Q(x) has integrating factor μ = exp(∫P dx) and solution yμ = ∫μQ dx + C.

For a second-order linear ODE, roots of the characteristic equation decide the solution: distinct real, repeated (multiply by x), or complex conjugate α ± iβ giving e^{αx}(C₁cos βx + C₂sin βx).

A function continuous on [a, b] attains its maximum and minimum there (extreme value theorem); differentiability implies continuity, not conversely.

Mean value theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = (f(b) − f(a))/(b − a).

Taylor series about a: f(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!; standard expansions include eˣ, sin x, cos x, ln(1 + x) and (1 + x)^α.

Convergence tests: Σ1/n^p converges iff p > 1; the ratio test is inconclusive when the limit of |a_{n+1}/a_n| equals 1.

Leibniz rule for limits of integration: d/dx ∫{a(x)}^{b(x)} f(x, t)dt = f(x, b)b′ − f(x, a)a′ + ∫{a}^{b} ∂f/∂x dt.

For a surface z = f(x, y), a stationary point with f_xx f_yy − f_xy² > 0 is an extremum (minimum if f_xx > 0), and a saddle if that quantity is negative.

Improper integral ∫₁^∞ dx/x^p converges iff p > 1; ∫₀¹ dx/x^p converges iff p < 1.

Rolle's theorem is the special case of the MVT where f(a) = f(b), giving f′(c) = 0 for some interior c.

A sequence converges iff it is Cauchy in a complete metric space such as ℝ; every convergent sequence is bounded, but bounded does not imply convergent.

Monotone convergence: a monotone increasing sequence bounded above converges to its supremum.

Bolzano–Weierstrass: every bounded sequence in ℝ has a convergent subsequence; a set is compact in ℝ iff closed and bounded (Heine–Borel).

Continuity on a compact set implies uniform continuity; uniform convergence of continuous functions preserves continuity of the limit.

Riemann integrability: a bounded f on [a, b] is integrable iff its set of discontinuities has measure zero (Lebesgue criterion).

A series Σaₙ converges absolutely ⇒ converges; absolute convergence permits rearrangement without changing the sum (Dirichlet).

Power series Σcₙ(x − a)ⁿ has radius R = 1/limsup|cₙ|^{1/n}, converging absolutely for |x − a| < R and diverging for |x − a| > R.

Supremum is the least upper bound; every nonempty subset of ℝ bounded above has a supremum (completeness axiom).

A normed space is complete if every Cauchy sequence converges in it; a complete normed space is a Banach space, and a complete inner-product space is a Hilbert space.

Hahn–Banach: a bounded linear functional on a subspace extends to the whole space with the same norm.

Open mapping theorem: a bounded linear surjection between Banach spaces is an open map; the closed graph theorem gives boundedness from a closed graph.

Uniform boundedness (Banach–Steinhaus): a pointwise-bounded family of bounded linear operators on a Banach space is uniformly bounded in norm.

Riesz representation: on a Hilbert space H, every bounded linear functional f has the form f(x) = ⟨x, y⟩ for a unique y, with ‖f‖ = ‖y‖.

Orthogonal projection onto a closed subspace of a Hilbert space is a bounded linear operator of norm 1 (when the subspace is nonzero).

The dual of ℓ^p is ℓ^q with 1/p + 1/q = 1 for 1 < p < ∞; the dual of ℓ¹ is ℓ^∞.

A compact operator maps bounded sets to relatively compact sets; the identity on an infinite-dimensional Banach space is never compact.

Dimension formula: dim(U + W) = dim U + dim W − dim(U ∩ W); for a linear map T: V → W, dim V = rank T + nullity T.

A matrix is diagonalisable iff the sum of geometric multiplicities of its eigenvalues equals n, equivalently each eigenvalue's algebraic multiplicity equals its geometric multiplicity.

Cayley–Hamilton: every square matrix satisfies its own characteristic polynomial, so Aⁿ can be reduced to lower powers.

A real matrix is orthogonally diagonalisable iff it is symmetric; a complex matrix is unitarily diagonalisable iff it is normal (AA* = A*A).

Positive definiteness: a symmetric matrix is positive definite iff all leading principal minors are positive (Sylvester's criterion) iff all eigenvalues are positive.

Singular value decomposition: A = UΣVᵀ with singular values σᵢ = √λᵢ(AAᵀ); rank equals the number of nonzero singular values.

Eigenvalues of a triangular or diagonal matrix are its diagonal entries; similar matrices share characteristic polynomial, trace, determinant and rank.

For a linear system Ax = b, a solution exists iff rank A = rank [A|b]; it is unique iff rank A = n.

Cauchy–Riemann equations uₓ = v_y, u_y = −vₓ are necessary for differentiability and sufficient when partials are continuous.

Cauchy's integral formula: f(z₀) = (1/2πi)∮_C f(z)/(z − z₀)dz for z₀ inside a simple closed contour C with f analytic on and inside C.

Liouville's theorem: a bounded entire function is constant; consequently every nonconstant polynomial has a root (fundamental theorem of algebra).

A pole of order m at z₀ has Res(f, z₀) = lim_{z→z₀} (1/(m−1)!) d^{m−1}/dz^{m−1}[(z − z₀)^m f(z)].

Argument principle: (1/2πi)∮ f′(z)/f(z)dz counts zeros minus poles of f inside the contour.

Conformal maps preserve angles; Möbius transformations (az + b)/(cz + d) with ad − bc ≠ 0 map circles and lines to circles and lines.

Taylor series converges inside the largest disc free of singularities; Laurent series is used in an annulus and its coefficients give residues.

An isolated singularity is removable if the Laurent series has no negative powers, a pole if finitely many, and essential if infinitely many.

Existence–uniqueness (Picard–Lindelöf): y′ = f(x, y), y(x₀) = y₀ has a unique local solution if f is continuous and Lipschitz in y near (x₀, y₀).

Linear independence of solutions on an interval is tested by the Wronskian W ≠ 0; for a second-order linear ODE, W(x) = W(x₀)exp(−∫P dx).

Method of undetermined coefficients applies when the forcing term is polynomial, exponential, sine or cosine; resonance requires multiplying the trial solution by x.

Variation of parameters for y″ + P y′ + Q y = R gives y_p = −y₁∫ y₂R/W dx + y₂∫ y₁R/W dx.

A homogeneous linear system x′ = Ax has solution x = e^{At}x₀; stability of the zero solution requires all eigenvalues of A to have negative real part.

For a 2×2 system, eigenvalues with negative real parts give a stable node or spiral; purely imaginary eigenvalues give a centre; opposite signs give a saddle.

Sturm–Liouville problems y″ + λy = 0 with boundary conditions have a discrete sequence of real eigenvalues λ₁ < λ₂ < … tending to infinity.

Bernoulli equation y′ + P y = Q yⁿ becomes linear via the substitution v = y^{1−n}.

First-order quasilinear PDE a(x, y, u)uₓ + b(x, y, u)u_y = c(x, y, u) is solved by the characteristic equations dx/a = dy/b = du/c.

Classification of Au_xx + Bu_xy + Cu_yy + … = 0 depends on B² − 4AC: elliptic if negative, parabolic if zero, hyperbolic if positive.

The one-dimensional wave equation u_tt = c²u_xx has d'Alembert solution u = f(x − ct) + g(x + ct), with characteristics x ± ct constant.

The heat equation u_t = αu_xx is parabolic and smooths data; the Laplace equation u_xx + u_yy = 0 is elliptic and satisfies the maximum principle.

Separation of variables for u_t = αu_xx on 0 < x < L with u(0,t) = u(L,t) = 0 gives u = Σ bₙ sin(nπx/L)exp(−αn²π²t/L²).

Method of images enforces Dirichlet boundary conditions for Laplace problems in half-planes or strips by reflecting singularities.

The Laplace equation with Dirichlet data has a unique solution (uniqueness follows from the maximum principle).

A first-order PDE is linear if it is linear in u and its partials, quasilinear if linear only in the highest derivatives, and fully nonlinear otherwise.

Newton–Raphson iteration x_{n+1} = xₙ − f(xₙ)/f′(xₙ) converges quadratically near a simple root; it fails when f′ is zero at the iterate.

Newton's forward difference formula uses Δf(x) = f(x + h) − f(x); the backward difference uses ∇f(x) = f(x) − f(x − h).

Trapezoidal rule error is −(b − a)h²f″(ξ)/12; Simpson's 1/3 rule error is −(b − a)h⁴f⁽⁴⁾(ξ)/180.

Gauss–Legendre quadrature with n nodes integrates exactly all polynomials of degree up to 2n − 1.

Gauss–Seidel and Jacobi iterations converge for strictly diagonally dominant matrices; the spectral radius of the iteration matrix must be less than 1.

LU decomposition without pivoting exists when all leading principal minors are nonzero; partial pivoting is needed for stability when a pivot is small.

Lagrange interpolation through n + 1 distinct nodes yields a unique polynomial of degree at most n; the error involves f⁽ⁿ⁺¹⁾(ξ)/(n + 1)! times Π(x − xᵢ).

Runge–Kutta methods are one-step methods; the classical fourth-order RK has local truncation error O(h⁵) and global error O(h⁴).

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B); events are independent iff P(A ∩ B) = P(A)P(B).

Expectation is linear: E[aX + bY] = aE[X] + bE[Y]; Var(X) = E[X²] − (E[X])² and Var(aX + b) = a²Var(X).

Binomial(n, p) has mean np and variance np(1 − p); Poisson(λ) has mean and variance both λ.

Exponential(λ) has mean 1/λ, variance 1/λ² and is memoryless; the normal N(μ, σ²) has mean μ and variance σ².

Central limit theorem: the standardised sum of i.i.d. variables with finite variance tends to N(0, 1) in distribution.

Maximum likelihood estimates maximise the log-likelihood; for a normal sample the MLEs are the sample mean and the biased variance with divisor n.

An estimator is unbiased if E[θ̂] = θ; the sample variance with divisor n − 1 is unbiased for σ².

Chebyshev's inequality: P(|X − μ| ≥ kσ) ≤ 1/k², and the correlation coefficient satisfies −1 ≤ ρ ≤ 1.

Standard form: maximise cᵀx subject to Ax = b, x ≥ 0; slack variables convert ≤ constraints to equalities.

The dual of max cᵀx, Ax ≤ b, x ≥ 0 is min bᵀy, Aᵀy ≥ c, y ≥ 0; weak duality gives cᵀx ≤ bᵀy.

Strong duality: if either problem has a finite optimum, both do, and the optimal objective values are equal.

Complementary slackness: xⱼ > 0 forces the j-th dual constraint tight, and yᵢ > 0 forces the i-th primal constraint tight.

A basic feasible solution corresponds to a vertex of the feasible region; the simplex method moves between adjacent vertices improving the objective.

A transportation problem is balanced when total supply equals total demand; the initial basic feasible solution needs m + n − 1 basic cells.

The Hungarian method solves assignment problems in polynomial time; dual variables uᵢ, vⱼ satisfy uᵢ + vⱼ = cᵢⱼ on assigned cells.

If the primal is unbounded, the dual is infeasible; if the primal is infeasible, the dual may be infeasible or unbounded.