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Linear algebra computations with full matrix support
Perform linear algebra operations with precision. Supports addition, multiplication, transpose, determinant, and inverse.
Matrix Properties
Matrix Decompositions
Eigenvalues & Eigenvectors
Subspaces
Addition (A + B): Matrices must have same dimensions. Each corresponding element is added.
Subtraction (A − B): Matrices must have same dimensions. Each element of B is subtracted from corresponding element of A.
Multiplication (A × B): Columns of A must equal rows of B. Result has dimensions rows(A) × cols(B).
Transpose (AT): Flips matrix over its diagonal. Result dimensions are cols(A) × rows(A).
Determinant det(A): Only for square matrices. Measures scaling factor of linear transformation. Zero determinant indicates singular matrix.
Inverse (A-1): Only for non-singular square matrices (det ≠ 0). A × A-1 = I (identity).
Trace tr(A): Sum of diagonal elements. Only for square matrices. Invariant under change of basis. Equal to sum of eigenvalues.
Rank rank(A): Number of linearly independent rows or columns. Indicates dimension of column/row space. Maximum rank is min(rows, cols).
Frobenius Norm ||A||F: Square root of sum of squared elements. Measures matrix magnitude. Always non-negative.
Condition Number κ(A): Ratio of largest to smallest singular value. Measures numerical stability. κ = 1 for orthogonal matrices. Large κ indicates ill-conditioned matrix.
QR Decomposition QR(A): Factorizes A = QR where Q is orthogonal (QTQ = I) and R is upper triangular. Used for solving least squares problems and eigenvalue algorithms.
LU Decomposition LU(A): Factorizes A = PLU where P is permutation, L is lower triangular, U is upper triangular. Efficient for solving linear systems. Only for square matrices.
Gram-Schmidt GS(A): Orthogonalizes column vectors of A using classical Gram-Schmidt process. Produces orthonormal basis for column space.
Eigenvalue λ(A): Computes dominant eigenvalue using power iteration method. For square matrices only. Converges to eigenvalue with largest absolute value. λv = Av for eigenvector v.
Nullspace null(A): Set of all vectors v where Av = 0. Basis vectors span the kernel. Dimension is n - rank(A) where n is number of columns.
Column Space col(A): Span of column vectors. Basis vectors are linearly independent columns. Dimension equals rank(A).
Keyboard Shortcuts: Tab/Shift+Tab to navigate cells, Arrow keys within grid, Enter to confirm input.
Full eigendecomposition via QR iteration. Displays characteristic polynomial, all eigenvalue/eigenvector pairs, and diagonalisation P D P⁻¹.
Eigenvalue λ: scalar where Av = λv for non-zero vector v.
Eigenvector v: direction preserved (only scaled) by A.
Char. poly p(λ): det(λI − A). Roots are eigenvalues.
Diagonalisation: A = PDP⁻¹ when P has full column rank.
Add, subtract, and scalar multiply matrices. Supports matrices of any size.
Compute matrix products with automatic dimension checking.
Calculate determinants using LU decomposition for square matrices.
Find inverses using Gauss-Jordan elimination. Handles singular matrices gracefully.
Transpose matrices to swap rows and columns.
Create identity, zero, ones, and random matrices.