Math Reference

Linear Algebra Terms and Calculation Guide

Learn vectors, matrices, affine and spatial geometry, lattices, linear systems, transformations, decompositions, least squares, and PCA through worked examples.

Point, plane, and normal vector

The shortest point-to-plane displacement is parallel to the plane's normal vector.

Lattice basis and fundamental region

Integer combinations of two basis vectors tile the plane with equal-area fundamental parallelograms.

97 matching terms

Objects and shapes

Scalar

Notationa

Meaning

A single numerical value used to scale vectors or matrices.

When to use it

Use scalars for weights, coefficients, learning rates, temperatures, and magnitudes.

Worked example

3[2, -1] = [6, -3].

Objects and shapes

Vector

Notationv ∈ ℝⁿ

Meaning

An ordered list of components that can represent direction, position, features, or state.

When to use it

Use vectors to represent coordinates, signals, features, embeddings, and model parameters.

Worked example

v = [3, 4] has two components.

Objects and shapes

Matrix

NotationA ∈ ℝᵐˣⁿ

Meaning

A rectangular array of numbers arranged in rows and columns.

When to use it

Use matrices to store datasets, linear systems, transformations, images, and weights.

Worked example

A = [[1, 2], [3, 4]].

Objects and shapes

Tensor

NotationT ∈ ℝᵈ¹ˣ⋯ˣᵈᵏ

Meaning

A multidimensional array that generalizes scalars, vectors, and matrices.

When to use it

Use tensors for batches, images, video, model activations, and multi-axis scientific data.

Worked example

A batch of 32 RGB images sized 224×224 has shape 32×3×224×224.

Objects and shapes

Shape

Notationm × n

Meaning

The ordered sizes of an array's axes.

When to use it

Check shapes before addition, multiplication, broadcasting, reshaping, and model input.

Caution

Most matrix multiplication errors come from incompatible inner dimensions.

Worked example

A 3×4 matrix has 3 rows and 4 columns.

Vector operations

Vector addition

Notationu + v

Meaning

Componentwise addition of vectors with the same dimension.

When to use it

Use it to combine displacements, forces, signals, updates, or feature contributions.

Worked example

[1, 2] + [3, -1] = [4, 1].

Vector operations

Scalar multiplication

Notationcv

Meaning

Multiplying every vector component by the same scalar.

When to use it

Use it to scale magnitude, reverse direction, or apply a weighted update.

Worked example

-2[3, 1] = [-6, -2].

Vector operations

Dot product

Notationu · v

Meaning

The sum of products of corresponding vector components, producing a scalar.

When to use it

Use it for similarity, projection, work, attention scores, and linear model output.

Worked example

[1, 2, 3] · [4, 0, -1] = 1.

Vector operations

Cross product

Notationu × v

Meaning

A three-dimensional vector perpendicular to two input vectors, with magnitude equal to their parallelogram area.

When to use it

Use it for surface normals, torque, orientation, and 3D geometry.

Caution

The standard cross product is specific to three dimensions, apart from a less common seven-dimensional analogue.

Worked example

[1,0,0] × [0,1,0] = [0,0,1].

Vector operations

Vector norm

Notation‖v‖

Meaning

A nonnegative measure of vector size that satisfies norm axioms.

When to use it

Use a norm to measure magnitude, distance, error, regularization, and convergence.

Worked example

For v=[3,4], ‖v‖₂=5.

Vector operations

Unit vector

Notationv/‖v‖

Meaning

A vector whose norm is 1.

When to use it

Use it to preserve direction while removing magnitude and to build orthonormal bases.

Worked example

[3,4]/5 = [0.6,0.8].

Vector operations

Euclidean distance

Notation‖u-v‖₂

Meaning

The straight-line distance between two points represented as vectors.

When to use it

Use it for geometry, nearest-neighbor search, clustering, and error measurement when scales are comparable.

Worked example

Distance from [1,1] to [4,5] is 5.

Vector operations

Cosine similarity

Notationu·v/(‖u‖‖v‖)

Meaning

The cosine of the angle between two nonzero vectors, measuring directional similarity.

When to use it

Use it to compare text embeddings or high-dimensional features when magnitude should matter less.

Caution

Cosine similarity is undefined for a zero vector and can hide meaningful magnitude differences.

Worked example

[1,0] and [2,0] have cosine similarity 1.

Vector operations

Orthogonal vectors

Notationu·v=0

Meaning

Vectors with zero dot product.

When to use it

Use orthogonality to separate independent directions, simplify projections, and construct stable bases.

Worked example

[1,2] · [2,-1] = 0, so the vectors are orthogonal.

Vector operations

Vector projection

Notationprojᵤ(v)

Meaning

The component of one vector that lies in the direction of another vector or subspace.

When to use it

Use it for decomposition, least squares, shadows, and removing a directional component.

Worked example

proj_[1,0]([3,4]) = [3,0].

Matrix operations

Matrix addition

NotationA+B

Meaning

Componentwise addition of matrices with the same shape.

When to use it

Use it to combine linear effects, residual updates, images, or accumulated data.

Worked example

[[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]].

Matrix operations

Matrix multiplication

NotationAB

Meaning

A row-by-column operation that composes linear transformations when inner dimensions match.

When to use it

Use it for coordinate transformations, neural-network layers, graph propagation, and solving systems.

Caution

Matrix multiplication is generally not commutative: AB can differ from BA or one order may be undefined.

Worked example

A₂ˣ₃B₃ˣ₄ produces C₂ˣ₄.

Matrix operations

Transpose

NotationAᵀ

Meaning

A matrix formed by exchanging rows and columns.

When to use it

Use it for dot products, covariance, normal equations, symmetry checks, and changing orientation.

Worked example

If A=[[1,2,3],[4,5,6]], then Aᵀ=[[1,4],[2,5],[3,6]].

Matrix operations

Identity matrix

NotationI

Meaning

A square matrix with ones on the main diagonal and zeros elsewhere.

When to use it

Use it as the multiplicative identity and to describe unchanged coordinates.

Worked example

AI = IA = A.

Matrix operations

Inverse matrix

NotationA⁻¹

Meaning

A matrix satisfying AA⁻¹=A⁻¹A=I for an invertible square matrix A.

When to use it

Use it conceptually to reverse a transformation and solve Ax=b.

Caution

Numerical software should usually solve Ax=b directly instead of explicitly computing A⁻¹.

Worked example

For A=[[2,0],[0,4]], A⁻¹=[[1/2,0],[0,1/4]].

Matrix operations

Determinant

Notationdet(A)

Meaning

A scalar for a square matrix that measures signed volume scaling and indicates invertibility.

When to use it

Use it to test singularity and analyze orientation or volume change under a transformation.

Worked example

det([[a,b],[c,d]]) = ad - bc.

Matrix operations

Trace

Notationtr(A)

Meaning

The sum of the main diagonal entries of a square matrix.

When to use it

Use it in eigenvalue identities, covariance analysis, matrix calculus, and optimization.

Worked example

tr([[2,1],[3,4]]) = 6.

Matrix operations

Matrix rank

Notationrank(A)

Meaning

The number of linearly independent rows or columns in a matrix.

When to use it

Use it to measure information dimension, determine solution structure, and detect redundant features.

Worked example

rank([[1,2],[2,4]]) = 1.

Matrix operations

Symmetric matrix

NotationA=Aᵀ

Meaning

A square matrix equal to its transpose.

When to use it

Use it for covariance, quadratic forms, undirected graphs, and real orthogonal eigendecomposition.

Worked example

[[2,3],[3,5]] is symmetric.

Matrix operations

Orthogonal matrix

NotationQᵀQ=I

Meaning

A real square matrix whose columns and rows form orthonormal sets.

When to use it

Use it for rotations, reflections, stable factorizations, and norm-preserving transforms.

Worked example

Q⁻¹ = Qᵀ for an orthogonal matrix.

Matrix operations

Diagonal matrix

NotationD

Meaning

A matrix whose entries outside the main diagonal are zero.

When to use it

Use it for independent scaling and efficient powers, inverses, and transformations.

Worked example

diag(2,3)^4 = diag(16,81).

Linear systems

System of linear equations

NotationAx=b

Meaning

A collection of linear equations that must be satisfied simultaneously.

When to use it

Use it for balancing, fitting, networks, circuits, constraints, and reconstruction.

Worked example

x+y=5 and 2x-y=1 give x=2, y=3.

Linear systems

Augmented matrix

Notation[A|b]

Meaning

A compact matrix representation that appends the right-hand side to a linear system's coefficient matrix.

When to use it

Use it to perform row reduction without repeatedly writing variables.

Worked example

x+2y=5, 3x-y=4 becomes [[1,2|5],[3,-1|4]].

Linear systems

Elementary row operation

Meaning

Swapping rows, scaling a row by a nonzero value, or adding a multiple of one row to another.

When to use it

Use these solution-preserving operations to simplify a linear system.

Worked example

R₂ ← R₂ - 3R₁.

Linear systems

Row echelon form

NotationREF

Meaning

A matrix form with pivots moving rightward and zeros below each pivot.

When to use it

Use it for back substitution, rank calculation, and identifying free variables.

Worked example

[[1,2,3],[0,1,4],[0,0,0]] is in row echelon form.

Linear systems

Reduced row echelon form

NotationRREF

Meaning

A row echelon form in which every pivot is 1 and is the only nonzero entry in its column.

When to use it

Use it to read unique solutions, free variables, rank, and null-space bases directly.

Worked example

RREF([[1,2],[2,4]]) = [[1,2],[0,0]].

Linear systems

Gaussian elimination

Meaning

Row operations that transform a system into row echelon form, followed by back substitution.

When to use it

Use it as a general method for solving moderate dense linear systems by hand or in software.

Worked example

Eliminate x from lower rows, then solve from the last pivot upward.

Linear systems

Gauss-Jordan elimination

Meaning

Row operations continued until the augmented matrix reaches reduced row echelon form.

When to use it

Use it when the complete solution structure or inverse is needed explicitly.

Worked example

Reduce [A|I] to [I|A⁻¹] when A is invertible.

Linear systems

Consistent system

Meaning

A linear system that has at least one solution.

When to use it

Use rank or row reduction to distinguish unique, infinite, and nonexistent solutions.

Worked example

A row [0 0 | 1] proves a system is inconsistent.

Vector spaces

Vector space

NotationV

Meaning

A set whose elements can be added and scaled while satisfying the vector-space axioms.

When to use it

Use it to treat coordinates, polynomials, functions, signals, and matrices within one framework.

Worked example

ℝ³ and the set of polynomials of degree at most 2 are vector spaces.

Vector spaces

Subspace

NotationW ⊆ V

Meaning

A subset of a vector space that is itself closed under vector addition and scalar multiplication.

When to use it

Use it to describe constrained directions, solution sets, feature spaces, and invariant structure.

Worked example

The plane x+y+z=0 through the origin is a subspace of ℝ³.

Vector spaces

Span

Notationspan{v₁,…,vₖ}

Meaning

The set of every linear combination of a given collection of vectors.

When to use it

Use it to describe all directions or outputs reachable from generators.

Worked example

span{[1,0],[0,1]} = ℝ².

Vector spaces

Linear independence

Meaning

A set of vectors is independent when only the all-zero coefficients produce the zero vector.

When to use it

Use it to detect redundant directions and select a basis.

Worked example

[1,0] and [0,1] are linearly independent.

Vector spaces

Basis

Meaning

A linearly independent set that spans a vector space.

When to use it

Use it to assign coordinates and represent every vector uniquely.

Worked example

{[1,0],[0,1]} is the standard basis of ℝ².

Vector spaces

Dimension

Notationdim(V)

Meaning

The number of vectors in any basis of a finite-dimensional vector space.

When to use it

Use it to measure independent degrees of freedom.

Worked example

dim(ℝ⁴)=4.

Vector spaces

Column space

NotationCol(A)

Meaning

The span of a matrix's columns, equal to all outputs Ax.

When to use it

Use it to determine whether Ax=b is solvable and what outputs a transformation can produce.

Worked example

Ax=b is solvable exactly when b belongs to Col(A).

Vector spaces

Null space

NotationNull(A)

Meaning

The set of vectors x satisfying Ax=0.

When to use it

Use it to describe invisible directions, homogeneous solutions, parameter redundancy, and constraints.

Worked example

If A=[1 2], then Null(A)=span{[-2,1]}.

Vector spaces

Rank-nullity theorem

Notationrank(A)+nullity(A)=n

Meaning

For a matrix with n columns, column-space dimension plus null-space dimension equals n.

When to use it

Use it to connect independent outputs with lost input degrees of freedom.

Worked example

A 3×5 matrix with rank 3 has nullity 2.

Linear transformations

Linear transformation

NotationT(u+v)=T(u)+T(v)

Meaning

A mapping that preserves vector addition and scalar multiplication.

When to use it

Use it to model rotation, scaling, projection, filtering, and linear layers.

Worked example

T([x,y])=[2x,y] scales the x direction by 2.

Linear transformations

Kernel

Notationker(T)

Meaning

The set of inputs mapped to the zero vector by a linear transformation.

When to use it

Use it to detect information lost by a transformation and test injectivity.

Worked example

T is one-to-one exactly when ker(T)={0}.

Linear transformations

Image

Notationim(T)

Meaning

The set of all outputs produced by a transformation.

When to use it

Use it to describe reachable outputs and test surjectivity.

Worked example

For matrix transformation T(x)=Ax, im(T)=Col(A).

Linear transformations

Change of basis

Meaning

Re-expressing the same vector or transformation using a different coordinate basis.

When to use it

Use it to align coordinates with geometry, simplify an operator, or move between local and global frames.

Worked example

If P contains new basis vectors as columns, then [v]new=P⁻¹v.

Spatial and affine geometry

Point

NotationP

Meaning

A location in an affine space that does not by itself have magnitude or direction.

When to use it

Use points for positions and subtract two points to obtain a displacement vector.

Caution

Adding two points is not intrinsically defined without choosing an origin or an affine combination.

Worked example

For P=(1,2) and Q=(4,6), the displacement Q-P=[3,4].

Spatial and affine geometry

Position vector

NotationOP

Meaning

A vector from a chosen origin O to a point P.

When to use it

Use it to represent points with coordinates after fixing an origin and basis.

Worked example

If O=(0,0,0) and P=(2,-1,3), then OP=[2,-1,3].

Spatial and affine geometry

Affine space

Meaning

A space of points in which differences of points are vectors but no origin is preferred.

When to use it

Use it to model geometry independently of an arbitrary coordinate origin.

Worked example

A translated plane is an affine space even when it does not pass through the origin.

Spatial and affine geometry

Affine combination

NotationΣαᵢPᵢ, Σαᵢ=1

Meaning

A weighted combination of points whose coefficients sum to 1.

When to use it

Use it for interpolation, centroids, barycentric coordinates, and affine transformations.

Worked example

The midpoint of P and Q is 0.5P+0.5Q.

Spatial and affine geometry

Parametric equation of a line

Notationx=p+tv

Meaning

A line represented by a point p and a nonzero direction vector v.

When to use it

Use it to generate line points and solve intersections with planes or other lines.

Worked example

Through p=(1,2) with v=[3,-1]: x(t)=(1+3t,2-t).

Spatial and affine geometry

Equation of a plane

Notationn·(x-p)=0

Meaning

A plane described by a point p and a nonzero normal vector n.

When to use it

Use it for classification boundaries, clipping, collision tests, and geometric constraints.

Worked example

With n=[1,2,3] and p=(1,0,0), the plane is x+2y+3z=1.

Spatial and affine geometry

Hyperplane

Notationw·x=b

Meaning

An affine subspace of dimension n-1 in an n-dimensional space.

When to use it

Use it as a decision boundary, constraint surface, or higher-dimensional plane.

Worked example

In ℝ⁴, w·x=b defines a three-dimensional hyperplane.

Spatial and affine geometry

Normal vector

Notationn

Meaning

A vector perpendicular to a line, plane, surface tangent space, or hyperplane.

When to use it

Use it to define planes, calculate distances, reflect vectors, and determine surface orientation.

Worked example

For 2x-y+3z=4, a normal vector is [2,-1,3].

Spatial and affine geometry

Line-plane intersection

Meaning

A point found by substituting a parametric line into a plane equation and solving for its parameter.

When to use it

Use it for ray casting, rendering, collision detection, and geometric construction.

Caution

If n·v=0, the line is parallel to the plane or lies entirely inside it.

Worked example

Substitute x=p+tv into n·x=d, then solve t=(d-n·p)/(n·v).

Spatial and affine geometry

Distance from a point to a line

Meaning

The length of the shortest perpendicular segment from a point to a line.

When to use it

Use it for nearest-path queries, fitting, collision margins, and geometric error.

Worked example

For line p+tv, distance(P,line)=‖(P-p)-projᵥ(P-p)‖.

Spatial and affine geometry

Distance from a point to a plane

Notation|n·P-d|/‖n‖

Meaning

The absolute signed plane equation at a point, normalized by the normal-vector length.

When to use it

Use it for margins, clipping, collision detection, and point-cloud processing.

Worked example

Distance from (1,2,3) to z=0 is 3.

Spatial and affine geometry

Projection onto a plane

Meaning

The closest point on a plane obtained by removing the normal component of a displacement.

When to use it

Use it to snap points to surfaces, resolve constraints, and decompose motion.

Worked example

For plane n·x=d, Pproj=P-((n·P-d)/‖n‖²)n.

Spatial and affine geometry

Reflection across a plane

Meaning

A transformation that reverses the normal component while preserving components parallel to the plane.

When to use it

Use it for mirror geometry, bounce directions, symmetry, and graphics.

Worked example

For a plane through the origin, vrefl=v-2projₙ(v).

Spatial and affine geometry

Barycentric coordinates

Notationα+β+γ=1

Meaning

Weights that express a point as an affine combination of simplex vertices.

When to use it

Use them for triangle interpolation, point-in-triangle tests, meshes, and finite elements.

Worked example

P=αA+βB+γC with α+β+γ=1.

Spatial and affine geometry

Area from a determinant

Notation|det([u v])|

Meaning

The absolute determinant of two planar edge vectors, equal to their parallelogram area.

When to use it

Use it for polygon area, orientation tests, Jacobians, and coordinate changes.

Worked example

u=[3,0], v=[1,2] give area |3×2-0×1|=6.

Spatial and affine geometry

Scalar triple product

Notationu·(v×w)

Meaning

A signed volume measure for the parallelepiped formed by three three-dimensional vectors.

When to use it

Use it for volume, coplanarity, and three-dimensional orientation tests.

Worked example

The volume is |u·(v×w)|.

Spatial and affine geometry

Orientation

Notationsign(det)

Meaning

A sign indicating the handedness or clockwise versus counterclockwise order of a basis or point sequence.

When to use it

Use it for polygon algorithms, winding, normals, and coordinate-system consistency.

Worked example

In 2D, det([B-A,C-A])>0 means A,B,C are counterclockwise.

Spatial and affine geometry

Homogeneous coordinates

Notation[x,y,z,w]

Meaning

Coordinates with an extra scale component that represent affine points and projective directions uniformly.

When to use it

Use them to combine translation, rotation, scaling, perspective, and projection in matrix form.

Caution

A homogeneous vector must be normalized carefully when its final component is nonzero; a zero final component represents a direction at infinity.

Worked example

The 2D point (x,y) becomes [x,y,1], while a direction becomes [vx,vy,0].

Lattice geometry

Lattice

NotationL=Bℤᵏ

Meaning

A discrete set of points formed by all integer combinations of linearly independent basis vectors.

When to use it

Use lattices in discrete geometry, coding, cryptography, optimization, and crystallography.

Worked example

For b₁=[2,0], b₂=[1,3], L={z₁b₁+z₂b₂ | z₁,z₂∈ℤ}.

Lattice geometry

Integer lattice

Notationℤⁿ

Meaning

The lattice of all n-dimensional vectors with integer coordinates.

When to use it

Use it as the standard coordinate lattice and a reference for sublattices and integer optimization.

Worked example

ℤ² contains every point (m,n) with m,n∈ℤ.

Lattice geometry

Lattice basis

NotationB=[b₁ … bₖ]

Meaning

A linearly independent set whose integer combinations generate a lattice.

When to use it

Use it to encode, enumerate, transform, and calculate properties of a lattice.

Caution

A lattice has infinitely many possible bases, often with very different vector lengths and angles.

Worked example

Columns [2,0] and [1,3] form a basis of a two-dimensional lattice.

Lattice geometry

Lattice rank

Notationrank(L)

Meaning

The number of vectors in a lattice basis, equal to the dimension of its real span.

When to use it

Use it to distinguish full-rank and lower-dimensional lattices inside an ambient space.

Worked example

The lattice generated by [1,0,0] and [0,1,0] has rank 2 in ℝ³.

Lattice geometry

Lattice point

NotationBz

Meaning

A point produced by multiplying a lattice basis matrix by an integer vector.

When to use it

Use it as the discrete candidate in closest-point, packing, coding, and integer-constraint problems.

Worked example

With B=[[2,1],[0,3]] and z=[2,-1], Bz=[3,-3].

Lattice geometry

Fundamental parallelepiped

NotationP(B)

Meaning

The half-open region formed by basis coefficients between 0 inclusive and 1 exclusive.

When to use it

Use it as one repeating cell that contains one representative of each coset modulo the lattice.

Worked example

P(B)={Bt | 0≤tᵢ<1}.

Lattice geometry

Lattice determinant

Notationdet(L)

Meaning

The volume of a fundamental region, also called the lattice covolume.

When to use it

Use it to measure lattice density and compare the spacing of full-rank lattices.

Worked example

For square basis B, det(L)=|det(B)|; B=[[2,1],[0,3]] gives 6.

Lattice geometry

Sublattice

NotationL'⊆L

Meaning

A subgroup of a lattice that is itself a lattice in the same real span or a lower-dimensional span.

When to use it

Use it to impose additional congruence conditions or compare nested discrete structures.

Worked example

2ℤ² is a sublattice of ℤ².

Lattice geometry

Lattice index

Notation[L:L']

Meaning

The number of cosets of a full-rank sublattice L' in L.

When to use it

Use it to measure how much sparser a sublattice is and relate determinants of nested lattices.

Worked example

[ℤ²:2ℤ²]=4 and det(2ℤ²)=4det(ℤ²).

Lattice geometry

Unimodular matrix

NotationU∈GLₙ(ℤ)

Meaning

An integer square matrix with determinant 1 or -1, whose inverse is also integral.

When to use it

Use it to change a lattice basis without changing the lattice itself.

Worked example

If B'=BU with det(U)=±1, then B and B' generate the same lattice.

Lattice geometry

Equivalent lattice bases

NotationB'=BU

Meaning

Two bases related by a unimodular integer matrix that generate exactly the same lattice.

When to use it

Use it to replace a long, skewed basis with a shorter and more orthogonal one.

Worked example

B and B'=B[[1,1],[0,1]] are equivalent bases.

Lattice geometry

Gram matrix of a lattice basis

NotationG=BᵀB

Meaning

A matrix of all pairwise inner products of basis vectors.

When to use it

Use it to calculate lengths, angles, volumes, and quadratic forms in basis coordinates.

Worked example

For integer vector z, ‖Bz‖²=zᵀGz.

Lattice geometry

Gram-Schmidt for lattice bases

Notationbᵢ*

Meaning

An orthogonalization used to analyze a lattice basis without generally producing another lattice basis.

When to use it

Use it to compute projection coefficients, basis quality, and LLL reduction steps.

Caution

Gram-Schmidt vectors are analytical auxiliaries and need not be lattice points.

Worked example

b₂*=b₂-μ₂₁b₁* with μ₂₁=⟨b₂,b₁*⟩/‖b₁*‖².

Lattice geometry

Orthogonality defect

Notation∏‖bᵢ‖/det(L)

Meaning

A measure of how far a full-rank basis is from being orthogonal.

When to use it

Use it to compare basis quality and anticipate numerical or enumeration difficulty.

Worked example

The defect equals 1 for an orthogonal basis and is greater than or equal to 1 otherwise.

Lattice geometry

Dual lattice

NotationL*

Meaning

The set of vectors having integer inner products with every vector in L.

When to use it

Use it in Fourier analysis, coding theory, reciprocal geometry, and transference bounds.

Worked example

For full-rank basis B, a dual basis is B⁻ᵀ.

Lattice geometry

Shortest vector problem

NotationSVP

Meaning

Finding a shortest nonzero vector in a lattice under a chosen norm.

When to use it

Use it to understand lattice geometry, reduction quality, and lattice-based cryptographic hardness.

Worked example

Find z≠0 minimizing ‖Bz‖.

Lattice geometry

Closest vector problem

NotationCVP

Meaning

Finding the lattice point closest to a target point.

When to use it

Use it for decoding, quantization, integer least squares, and lattice-based security analysis.

Worked example

Find z minimizing ‖Bz-t‖.

Lattice geometry

Successive minima

Notationλᵢ(L)

Meaning

Radii needed to contain increasing numbers of linearly independent lattice vectors.

When to use it

Use them to describe lattice shape beyond only the single shortest vector.

Worked example

λ₁(L) is the shortest-vector length, while λₖ(L) reaches k independent vectors.

Lattice geometry

Minkowski's convex body theorem

Meaning

A volume condition ensuring that a symmetric convex body contains a nonzero lattice point.

When to use it

Use it to prove bounds on short lattice vectors and results in algebraic number theory.

Worked example

A sufficiently large origin-symmetric convex body must contain a nonzero point of L.

Lattice geometry

Lattice sphere packing

Meaning

Placing equal nonoverlapping spheres at lattice points and measuring the occupied-space fraction.

When to use it

Use it in coding theory, communications, discrete geometry, and high-dimensional optimization.

Worked example

The packing radius is half the shortest nonzero lattice-vector length.

Lattice geometry

Voronoi cell of a lattice

Meaning

The region of points at least as close to one lattice point as to every other lattice point.

When to use it

Use it to understand nearest-lattice-point decoding and the geometric shape of CVP regions.

Worked example

The Voronoi cell around 0 tiles space by lattice translations.

Lattice geometry

Lattice basis reduction

Meaning

Replacing a lattice basis by an equivalent basis with shorter and more nearly orthogonal vectors.

When to use it

Use it to improve enumeration, integer-relation search, cryptanalysis, and numerical behavior.

Worked example

A reduced basis generates the same lattice but exposes its geometry more clearly.

Lattice geometry

LLL algorithm

NotationLLL

Meaning

A polynomial-time algorithm that produces a basis satisfying size-reduction and Lovász conditions.

When to use it

Use it for practical approximate short vectors, polynomial factoring, cryptanalysis, and integer relations.

Caution

LLL gives a quality guarantee for an approximate short vector, not necessarily the exact SVP solution.

Worked example

LLL repeatedly size-reduces Gram-Schmidt coefficients and swaps basis vectors when the Lovász condition fails.

Eigenvalues and decompositions

Eigenvalue

NotationAv=λv

Meaning

A scalar λ by which a linear transformation scales a nonzero eigenvector without changing its direction.

When to use it

Use eigenvalues to study stability, long-term dynamics, covariance, graphs, and differential equations.

Worked example

For A=diag(2,3), the eigenvalues are 2 and 3.

Eigenvalues and decompositions

Eigenvector

NotationAv=λv, v≠0

Meaning

A nonzero direction preserved by a linear transformation up to scalar scaling.

When to use it

Use it to identify natural axes, dominant modes, steady states, and principal directions.

Worked example

For A=diag(2,3), [1,0] is an eigenvector for λ=2.

Eigenvalues and decompositions

Characteristic polynomial

Notationdet(A-λI)

Meaning

A polynomial whose roots are the eigenvalues of a square matrix.

When to use it

Use it for symbolic eigenvalue calculations and theoretical analysis of small matrices.

Worked example

For A=[[2,0],[0,3]], det(A-λI)=(2-λ)(3-λ).

Eigenvalues and decompositions

Diagonalization

NotationA=PDP⁻¹

Meaning

Representing a matrix using a diagonal matrix of eigenvalues and a basis of eigenvectors.

When to use it

Use it to simplify matrix powers, recurrences, and linear dynamical systems.

Caution

Not every square matrix has enough independent eigenvectors to be diagonalizable.

Worked example

A^k=PD^kP⁻¹ when A is diagonalizable.

Eigenvalues and decompositions

LU decomposition

NotationPA=LU

Meaning

Factoring a matrix into lower- and upper-triangular factors, sometimes with row permutation.

When to use it

Use it to solve several systems with the same coefficient matrix efficiently.

Worked example

Factor PA=LU, then solve Ly=Pb and Ux=y.

Eigenvalues and decompositions

QR decomposition

NotationA=QR

Meaning

Factoring a matrix into an orthogonal matrix Q and an upper-triangular matrix R.

When to use it

Use it for numerically stable least squares, orthonormal bases, and eigenvalue algorithms.

Worked example

Solve least squares by Rx=Qᵀb after A=QR.

Eigenvalues and decompositions

Singular value decomposition

NotationA=UΣVᵀ

Meaning

A factorization of any matrix into orthogonal singular-vector matrices and nonnegative singular values.

When to use it

Use it for compression, denoising, pseudoinverses, low-rank approximation, and latent structure.

Caution

Small singular values can amplify noise when used in an inverse or pseudoinverse.

Worked example

Keeping the largest k singular values gives the best rank-k approximation in the 2-norm and Frobenius norm.

Eigenvalues and decompositions

Least squares

Notationmin ‖Ax-b‖₂

Meaning

Finding parameters that minimize the squared residual when a linear system has no exact solution or is overdetermined.

When to use it

Use it for regression, calibration, reconstruction, and fitting noisy measurements.

Worked example

Fit y≈mx+c by minimizing the sum of squared vertical residuals.

Eigenvalues and decompositions

Principal component analysis

NotationX≈UₖΣₖVₖᵀ

Meaning

A dimensionality-reduction method that finds orthogonal directions of greatest variance in centered data.

When to use it

Use it to visualize, compress, denoise, or summarize correlated numerical features.

Caution

PCA is sensitive to feature scale, outliers, and the assumption that high variance is informative.

Worked example

Center X, compute its SVD, and project onto the first k right singular vectors.