Matrices & vectors
Matrix arithmetic, determinants, inverses and the classical decompositions (LU, QR, SVD, eigenvalues). 22 tools, including Adjacency matrix calculator, Angle Between Two Vectors Calculator, Cholesky decomposition calculator and Column Space Calculator (basis of Col(A), rank, pivot columns). All free, with no account and nothing to install.
Adjacency matrix calculatorTurn any graph into its adjacency matrix. Paste an edge list (A-B), an adjacency list (A: B C D) or a binary matrix — the tool auto-detects the format, builds the labelled adjacency matrix, and reports the degree sequence, in/out-degrees for directed graphs, and the vertex and edge counts.Angle Between Two Vectors Calculatorθ = arccos(a·b / (|a||b|)) in 2D or 3D, in degrees and radians, with the dot product, both magnitudes, the projections and a parallel / perpendicular / acute / obtuse verdict.Cholesky decomposition calculatorFactor a symmetric positive-definite matrix A into L·Lᵀ, where L is lower triangular — the Cholesky decomposition. Enter A on an editable square grid and it returns L, Lᵀ and the product L·Lᵀ; if the matrix is not symmetric or not positive-definite, it tells you which condition failed. Twice as fast as LU, Cholesky is used in optimization, Kalman filters and Monte-Carlo simulation.Column Space Calculator (basis of Col(A), rank, pivot columns)A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.Cramer's Rule CalculatorSolves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none.Cross Product Calculatora × b in 3D with the determinant expansion shown, the magnitude |a||b|sin θ, the parallelogram and triangle areas, the unit normal and the right-hand rule.Dot Product Calculatora·b in 2 to 10 dimensions with the term-by-term sum, both magnitudes as exact radicals, the angle in degrees and radians, the projections, and a figure. Parallel and orthogonal are decided in exact fractions, so (1,1,1)·(2,2,2) returns 0°, not NaN.Eigenvalue and Eigenvector Calculator (2×2 and 3×3, complex supported)Characteristic polynomial, exact eigenvalues — rational, surd like (1±√5)/2, or complex like ±i for a rotation — with eigenvectors, algebraic and geometric multiplicities, and a clear verdict on diagonalizability.Gram-Schmidt calculator (orthonormal basis)Turn a set of vectors into an orthonormal basis with the Gram-Schmidt process: each vector is made orthogonal to the previous ones and then normalized to unit length. Enter one vector per row on an editable grid and it returns the orthonormal basis, dropping any linearly dependent vectors. Orthonormal bases are the foundation of QR factorization, Fourier analysis and projections.Matrix CalculatorAdd, subtract and multiply matrices, scale, transpose, and find the determinant and inverse of A.2×2 matrix determinant calculatorCompute the determinant of a 2×2 matrix (ad − bc).LU Decomposition CalculatorFactor a square matrix into a lower L and upper U triangular matrix (Doolittle method).Matrix Power Calculator (Aⁿ, n from −10 to 20)Raise a 1×1 to 5×5 matrix to any power from −10 to 20 in exact fractions: A⁰ = I, binary exponentiation for positive n, the exact inverse for negative n, and an honest det(A) = 0 when no inverse exists.Matrix Rank CalculatorThe rank of a matrix by row reduction — the number of linearly independent rows.Matrix Trace CalculatorThe trace of a square matrix — the sum of its main-diagonal entries.Null Space Calculator (basis of Nul(A), nullity, parametric solution)Solve Ax = 0 for any matrix up to 8×8: a basis of the kernel in exact fractions and in cleared integers, the free variables, the parametric solution, the rank–nullity check and A·v = 0 verified exactly.QR decomposition calculatorFactor a matrix A into Q·R, where Q has orthonormal columns and R is upper triangular, using the modified Gram-Schmidt process. Enter A on an editable grid (rows ≥ columns), and it returns Q, R and the product Q·R so you can check the factorization at a glance. QR is the workhorse behind least-squares regression and eigenvalue algorithms.RREF Calculator (Reduced Row Echelon Form)Gauss-Jordan elimination up to 10×10 in exact rational arithmetic, with every row operation written out as R₂ ← R₂ − 2R₁ and the matrix redrawn after each one. Augmented mode classifies the system and gives the solution set.SVD calculator (singular value decomposition)Compute the singular value decomposition A = U·Σ·Vᵀ of any matrix. The singular values are the square roots of the eigenvalues of AᵀA, found with the Jacobi eigenvalue algorithm; U and V are orthogonal. Enter A on an editable grid (up to 6×6) and it returns the singular values, U, Σ, Vᵀ and the reconstruction U·Σ·Vᵀ. SVD underpins PCA, image compression, recommender systems and the pseudo-inverse.Tensor Product Calculator (Kronecker product A ⊗ B)The Kronecker product of two matrices of any shape, block by block, in exact fractions or 6 / 10 significant digits — with the (m·p) × (n·q) size rule and the identities on rank, transpose and determinant.Unit Vector Calculatorv̂ = v / |v| in 2 to 10 dimensions, with |v| as a simplified radical, each component exact (−3,6,−2 gives −3/7, 6/7, −2/7), the share each component takes of |v|², the direction cosines and a unit-circle figure.Vector magnitude calculatorCompute the magnitude (length) of a 3D vector from its components.