math concept
67 topics use this
Math concept
Linear Algebra
Core equation
$$A\mathbf{v} = \lambda\mathbf{v}$$
Linear algebra is the mathematics of vectors, matrices, and linear transformations. It is the language of data science, physics, and engineering — underpinning regression, neural networks, quantum mechanics, and signal processing.
Vector spaces
A vector space over $\mathbb{R}$ is a set $V$ with addition and scalar multiplication satisfying eight axioms (closure, associativity, identity elements, inverses, distributivity).
Key subspaces of a matrix $A \in \mathbb{R}^{m \times n}$:
- Column space $\text{col}(A)$: set of all $A\mathbf{x}$ — relevant for OLS (projection)
- Null space $\text{null}(A)$: set of all $\mathbf{x}$ with $A\mathbf{x}=\mathbf{0}$ — relevant for identifiability
- Row space $\text{row}(A)$: complement of the null space
Matrix decompositions
| Decomposition | Form | Applications |
|---|---|---|
| Eigendecomposition | $A = Q\Lambda Q^{-1}$ | PCA, spectral clustering |
| SVD | $A = U\Sigma V^\top$ | Dimensionality reduction, pseudoinverse |
| QR | $A = QR$ | Numerically stable OLS |
| Cholesky | $A = LL^\top$ | Gaussian sampling, GP regression |
| LU | $A = LU$ | Linear system solving |
Fields that use this concept
Life sciences
Bioinformatics
Genome-Wide Association Studies
Statistical methods for linking genetic variants to traits or diseases across the entire genome.
Network Biology
Graph-theoretic analysis of protein-protein interaction networks, hub genes, and community structure.
Protein Structure Prediction
Energy minimization, force fields, and deep learning approaches for predicting 3D protein structure from sequence.
Single-Cell RNA Sequencing
Dimensionality reduction, clustering, and trajectory inference for high-dimensional single-cell transcriptomic data.
Life sciences
Biostatistics
Physical sciences
Computational chemistry
Coupled Cluster Theory
The gold-standard wavefunction method based on an exponential cluster operator that systematically captures electron correlation.
Hartree-Fock Theory
The foundational mean-field method for solving the electronic Schrödinger equation using antisymmetrised orbital products.
Molecular Mechanics
Classical force field models that treat atoms as point masses connected by springs to simulate large biomolecular systems.
Perturbation Theory in Quantum Chemistry
Systematic expansion of energies and wavefunctions in powers of a small perturbation, including the Møller-Plesset treatment of electron correlation.
Engineering & CS
Cryptography
AES Block Cipher
The Advanced Encryption Standard, a symmetric block cipher operating on 128-bit blocks with Galois field arithmetic.
Lattice-Based Cryptography
Post-quantum cryptographic constructions built on the computational hardness of lattice problems like LWE and SVP.
Secret Sharing Schemes
Methods for distributing a secret among multiple parties so that only authorized subsets can reconstruct it.
Finance & economics
Econometrics
ARIMA & Time Series
Modelling autocorrelated sequences. Connects deeply to signal processing and Fourier analysis.
Cointegration
Long-run equilibrium between non-stationary time series. Bridges econometrics and finance.
Difference-in-Differences
Estimating treatment effects using panel data. The go-to design for policy evaluation.
Generalized Method of Moments
A general estimation framework that exploits population moment conditions to identify and estimate model parameters with minimal distributional assumptions.
Instrumental Variables
Causal inference when OLS is biased by endogeneity. Uses a third variable to isolate exogenous variation.
OLS Regression
Estimating linear relationships by minimising squared residuals. The workhorse of econometrics.
Panel Data and Fixed Effects
Methods for estimating causal effects from panel data by exploiting within-unit variation to control for time-invariant unobserved heterogeneity.
Vector Autoregression
A multivariate time-series model in which each variable is regressed on its own lags and the lags of all other variables in the system.
Earth sciences
Geophysics
Earthquake Seismology
Moment tensors, seismic moment, and the Gutenberg-Richter law characterise earthquake sources and their statistical recurrence.
Electrical Methods
Resistivity surveys and IP measurements map subsurface conductivity by injecting current and measuring potential differences.
Geodesy
Reference ellipsoids, geoid undulations, GPS trilateration, and InSAR measure Earth's shape, gravity field, and crustal deformation.
Geostatistics
Variograms, kriging, and sequential simulation quantify spatial uncertainty in subsurface property estimation.
Gravity Methods
Bouguer anomalies and gravitational potential inversion reveal subsurface density contrasts.
Magnetic Methods
Total-field anomaly maps, Euler deconvolution, and susceptibility inversion image subsurface magnetic sources.
Seismic Tomography
Ray-path integrals and least-squares inversion map 3D velocity structure inside the Earth.
Engineering & CS
Machine learning
Backpropagation
Efficient computation of gradients in neural networks via the chain rule of calculus.
Gaussian Processes
A non-parametric Bayesian approach that places a prior directly over functions.
Gradient Descent
Iterative optimisation by following the steepest downhill direction. The engine of modern ML.
Linear Regression
The simplest supervised learning model — mathematically identical to econometric OLS.
Neural Networks
Universal function approximators trained by backpropagation, forming the foundation of modern deep learning.
Principal Component Analysis
Linear dimensionality reduction by projecting data onto directions of maximum variance via eigendecomposition of the covariance matrix.
Support Vector Machines
Maximum-margin classifiers that find the optimal separating hyperplane using convex quadratic programming and the kernel trick.
Earth sciences
Meteorology
Engineering & CS
Operations research
Dynamic Programming
Solving complex optimisation problems by breaking them into overlapping subproblems. Bellman's principle of optimality is the key insight.
Linear Programming
Optimising a linear objective over a polytope. The simplex algorithm and interior-point methods are the two main solution approaches.
Confirmatory Factor Analysis
A theory-driven factor model in which loadings and factor correlations are constrained a priori and tested against data.
Factor Analysis
Dimensionality-reduction technique decomposing observed variables into latent common factors and unique error terms.
Structural Equation Modeling
A framework combining a measurement model and a structural model to test hypothesized relationships among latent variables.
Life sciences
Quant ecology
Finance & economics
Quant finance
Black–Scholes Model
The foundational option pricing model. Derives a fair price from stochastic calculus and no-arbitrage.
Copulas
Functions that model the dependence structure between random variables independently of their marginal distributions.
Portfolio Optimization
Finding the allocation of capital across assets that maximises return for a given risk level. Markowitz's mean-variance framework is the foundation.
Life sciences
Quant genetics
The Animal Model
BLUP breeding value estimation using pedigree relationships and mixed model equations.
Epistasis
Gene-gene interactions and their contribution to phenotypic variance and GWAS inflation.
Genome-Wide Association Studies
Statistical methods for mapping SNP associations to complex traits across the genome.
Heritability
Partitioning phenotypic variance into genetic and environmental components.
Polygenic Scores
Aggregating genome-wide SNP effects into individual-level genetic prediction scores.
Quantitative Trait Loci
Statistical mapping of genomic regions that contribute to variation in quantitative traits.
Physical sciences
Quantum computing
Density Matrices
The mathematical framework for describing both pure and mixed quantum states, open systems, and quantum channels.
Grover's Algorithm
A quantum search algorithm achieving quadratic speedup over classical exhaustive search via amplitude amplification.
Quantum Entanglement
The non-classical correlations between quantum systems that cannot be explained by any local hidden-variable theory.
Quantum Error Correction
Techniques for protecting quantum information from decoherence and gate errors using redundant encoding and syndrome measurements.
Quantum Gates
The fundamental building blocks of quantum circuits, implementing unitary transformations on qubit states.
Quantum Phase Estimation
A core quantum subroutine that extracts eigenphases of unitary operators, underpinning Shor's algorithm and quantum chemistry simulations.
Shor's Algorithm
A quantum algorithm for integer factorisation running in polynomial time, breaking RSA encryption.
Variational Quantum Eigensolver
A hybrid quantum-classical algorithm for approximating ground-state energies of quantum systems using parameterised circuits.
Engineering & CS
Robotics
Kalman Filter SLAM
Simultaneous localization and mapping using the Extended Kalman Filter to jointly estimate robot pose and landmark positions.
Robot Kinematics
Forward and inverse kinematics for serial manipulators using homogeneous transforms, DH parameters, and the geometric Jacobian.
Robot Dynamics
Equations of motion for rigid-body robot manipulators derived via Lagrangian mechanics and computed efficiently with the Newton-Euler algorithm.
Visual Odometry
Estimating ego-motion from camera images by tracking visual features across frames using epipolar geometry and PnP solvers.
Engineering & CS
Signal processing
Adaptive Filtering
Filters that update their coefficients online to minimize a cost function, enabling noise cancellation, echo suppression, and channel equalization.
Compressed Sensing
A framework for recovering sparse signals from far fewer measurements than classical sampling theory requires.
Fourier Transform
Decomposing any signal into its constituent frequencies. One of the most powerful tools in all of mathematics.
Kalman Filter
The optimal recursive estimator for linear Gaussian state-space models, fusing predictions with noisy observations.
Wavelet Transform
A time-frequency analysis tool that overcomes the fixed resolution of the STFT by using dilated and translated basis functions.