📖 Matrix Glossary
Linear Algebra Glossary
Every linear algebra and matrix term, clearly defined with formulas and examples. From adjugate to zero matrix.
A
Adjugate (Adjoint) Matrix adj(A)
The transpose of the cofactor matrix of A. Used to compute the inverse: A⁻¹ = adj(A)/det(A).
$$A^{-1} = \frac{1}{\det(A)}\,\text{adj}(A)$$
Augmented Matrix
A matrix formed by appending columns of one matrix to another. For system Ax=b, the augmented matrix is [A|b]. Used in Gaussian elimination to solve linear systems.
B
Basis
A set of linearly independent vectors that span a vector space. Every vector in the space can be written uniquely as a linear combination of basis vectors.
C
Characteristic Polynomial
The polynomial p(λ) = det(A − λI). Its roots are the eigenvalues of A. For an n×n matrix it has degree n.
$$p(\lambda)=\det(A-\lambda I)=0$$
Cofactor
The cofactor Cᵢⱼ of element aᵢⱼ is (−1)^(i+j) × Mᵢⱼ where Mᵢⱼ is the minor (determinant of matrix with row i and column j removed).
$$C_{ij} = (-1)^{i+j} M_{ij}$$
Column Space (Range)
The set of all vectors that can be written as Ax for some vector x. Equivalently, the span of the columns of A. dim(Col A) = rank(A).
Condition Number
A measure of how sensitive a matrix equation Ax=b is to small perturbations. κ(A) = σ_max/σ_min (ratio of largest to smallest singular value). Large condition number → numerically ill-conditioned.
D
Determinant det(A) or |A|
A scalar value computed from a square matrix encoding its invertibility, volume scaling factor, and other properties. det(A) ≠ 0 iff A is invertible.
Diagonal Matrix
A square matrix where all entries outside the main diagonal are zero. Diagonal matrices are easy to raise to powers: D^n = diag(d₁^n,...,dₙ^n).
Diagonalization
A matrix A is diagonalizable if A = PDP⁻¹ where D is diagonal and P's columns are eigenvectors of A. Not all matrices are diagonalizable.
E
Eigenvalue & Eigenvector
A scalar λ and non-zero vector v such that Av = λv. The eigenvector v keeps its direction (or reverses) when multiplied by A; λ is the scaling factor.
$$A\mathbf{v}=\lambda\mathbf{v}$$
Eigenspace
For eigenvalue λ, the null space of (A − λI): all eigenvectors corresponding to λ plus the zero vector. Eigenspace = ker(A − λI).
F
Frobenius Norm
The square root of the sum of squares of all entries: ‖A‖_F = √(Σᵢⱼ aᵢⱼ²). Equal to √(sum of squares of singular values).
$$\|A\|_F = \sqrt{\sum_{i,j} a_{ij}^2} = \sqrt{\text{tr}(A^T A)}$$
G
Gram-Schmidt Process
An algorithm for orthogonalizing a set of linearly independent vectors, producing an orthonormal basis for the same subspace. Foundation of QR decomposition.
I
Identity Matrix I or Iₙ
The square matrix with 1s on the main diagonal and 0s elsewhere. AI = IA = A for any compatible matrix A.
Inverse Matrix A⁻¹
For square non-singular A, the unique matrix such that AA⁻¹ = A⁻¹A = I. A matrix is invertible iff det(A) ≠ 0.
L
Linear Combination
A sum c₁v₁ + c₂v₂ + ... + cₙvₙ where cᵢ are scalars and vᵢ are vectors. The set of all linear combinations is the span.
Linear Independence
Vectors v₁,...,vₙ are linearly independent if c₁v₁+...+cₙvₙ = 0 implies all cᵢ = 0. Equivalently, no vector is a linear combination of the others.
LU Decomposition
Factorization A = LU where L is lower triangular (1s on diagonal) and U is upper triangular. Used for efficient linear system solving and determinant computation.
N
Null Space (Kernel) ker(A)
The set of all vectors x such that Ax = 0. dim(null space) = nullity(A) = n − rank(A) (rank-nullity theorem).
Nullity
The dimension of the null space of A. Nullity = n − rank(A) by the rank-nullity theorem.
O
Orthogonal Matrix
A square matrix Q where Q^T Q = I (equivalently Q^T = Q^{-1}). Columns are orthonormal. det(Q) = ±1. Orthogonal matrices preserve lengths and angles.
Orthonormal Set
A set of vectors that are pairwise orthogonal and each has unit norm: ⟨vᵢ, vⱼ⟩ = δᵢⱼ (Kronecker delta).
P
Positive Definite Matrix
A symmetric matrix A where xᵀAx > 0 for all non-zero x. Equivalently, all eigenvalues are positive, all leading minors are positive.
Q
QR Decomposition
Factorization A = QR where Q is orthogonal and R is upper triangular. Computed via Gram-Schmidt or Householder transformations. Used in eigenvalue algorithms and least squares.
R
Rank
The dimension of the column space (= row space) of a matrix. Equals the number of non-zero rows in RREF. A square n×n matrix has full rank iff det(A) ≠ 0.
RREF — Reduced Row Echelon Form
A unique matrix form obtained via row operations where: each non-zero row's leading entry is 1 (pivot), each pivot is to the right of pivots above, all entries above and below each pivot are 0, and zero rows are at the bottom.
Row Space
The span of the rows of a matrix. dim(Row A) = rank(A) = dim(Col A). Row space ⊥ null space.
S
Singular Matrix
A square matrix with det(A) = 0. A singular matrix has no inverse. Its rows/columns are linearly dependent.
SVD — Singular Value Decomposition
Factorization A = UΣVᵀ. U and V are orthogonal; Σ is diagonal with non-negative singular values σ₁ ≥ ... ≥ σₙ ≥ 0. Generalizes eigendecomposition to non-square matrices.
Span
The set of all linear combinations of a set of vectors. span{v₁,...,vₙ} is the smallest subspace containing all vᵢ.
Symmetric Matrix
A matrix equal to its transpose: A = Aᵀ. Symmetric matrices always have real eigenvalues and an orthonormal set of eigenvectors (Spectral Theorem).
T
Trace tr(A)
The sum of the diagonal entries of a square matrix. tr(A) = Σᵢ aᵢᵢ = sum of eigenvalues. tr(AB) = tr(BA).
Transpose Aᵀ
Matrix formed by swapping rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. If A is m×n then Aᵀ is n×m.
Z
Zero Matrix O
The matrix of all zeros. Acts as the additive identity: A + O = A. det(O) = 0.