A curated archive of 54 self-contained mathematical derivations, theorem proofs, and algorithmic designs.
Linear Algebra & Matrix Theory
- Non-invertibility of 2x2 Matrix Pencils
- Diagonalizability of the Transpose Map across Field Characteristics
- Invariance of Characteristic Polynomials Under Conjugation
- Eigenvalues of Polynomials
- Invariance of Eigenspaces Under Transposition
- Invertibility of the Linear Functionals of a Dual Basis
- Diagonalizability of Reflection
- Basis and Polynomials of a Cyclic Subspace
- T-Invariant Subspaces and Cayley-Hamilton Theorem via Cyclic Subspaces
- Simultaneous Diagonalization of Commuting Linear Transformations
- Minimal Polynomial of Invertible Linear Transformations and Inverses
- Diagonalizability of the Transpose Map across Field Characteristics
- Jordan Canonical Form of the Transpose Map over Finite Fields
- Enumerating Conjugacy Classes via Splitting Characteristic Polynomials
- Computing the Jordan Canonical Form and Modal Matrix of a 4x4 Matrix
- Frobenius Inner Product on Matrix Spaces
- Parallelogram Law and Polarization Identity
- Orthogonal Projection Operators via Orthonormal Bases
- Best Approximation Theorem for Orthogonal Projections
- Riesz Representation Theorem and Dual Space Isomorphisms
- Computing Riesz Representatives on Polynomial Spaces
- Algebraic Properties of Adjoint Operators
- Self-Adjoint Operators and Characterization of Isometries
Real & Functional Analysis
- Fixed Points on Compact Spaces
- Geometric Series with Matrix Norm
- Cauchy Sequences and Functions on the Closure
- Completeness of Normed Vector Spaces via Absolutely Converging Series
- Cauchy Condensation Test and Generalized Bertrand Series
- Bertrand’s Convergence Test and Telescoping Series Bounds
- Hölder’s Inequality for Sequence Spaces and Absolute Convergence
- Summation by Parts and Abel’s Theorem for Power Series
- Domain Additivity of Riemann-Stieltjes Integrals
- Riemann-Stieltjes Integration Against Step Functions
- Reduction of Riemann-Stieltjes Integrals to Riemann Integrals
- Evaluation of Riemann-Stieltjes Integrals with Infinite Jump Discontinuities
- Hölder’s Inequality for Riemann-Stieltjes Integrals
- Proving Convexity of Functions Using Taylor’s Theorem
- Binomial Series Convergence via Taylor’s Integral Remainder
- Integral Comparison Test for the Bertrand Series
- Integral Representations of the Riemann Zeta Function
Ordinary Differential Equations
- Picard Iteration and the Cauchy-Lipschitz Theorem
- Maximal Interval of Existence for Ordinary Differential Equations
- Homogenous First-Order Linear Systems
- Homogenous Defective First-Order Linear Systems
- Solving Non-Exact Differential Equations
- Solving First-Order Linear Equations
Algorithms & Data Structures
- Overlap Detection with Sweep Line Algorithm
- Linear-Time Red-Black Tree Construction from Sorted List
- Upper Bounds on Red Nodes in a Red-Black Tree
- Non-Lazy Deletion in Tournament Trees
- Designing Deque for O(1) Amortized Time
- Proving Optimality and Designing Huffman Encoding Algorithm
- The Cut Lemma
- Max-Flow Min-Cut Theorem