Constructs a matrix from row arrays -- exactly 2x2 or 3x3 dispatches to the fixed-size fast path, anything else (including non-square) uses the general matrix type. A power A^k works for k from 1 to 10000, while n^2 * k * d is at most 30000, where n is the matrix's size and d the digits of its largest entry.
rows (array[]) -- one or more equal-length row arraysThe determinant of a matrix.
m (matrix) -- The trace (sum of diagonal entries) of a matrix.
m (matrix) -- The inverse of a matrix, computed exactly (fractions, not floats). Returns explanatory text ('singular, no inverse') when the matrix has no inverse.
m (matrix) -- The transpose of a matrix.
m (matrix) -- Adds two matrices of the same shape.
a (matrix) -- b (matrix) -- same shape as aSubtracts b from a, for two matrices of the same shape.
a (matrix) -- b (matrix) -- same shape as aMultiplies two compatibly-shaped matrices.
a (matrix) -- b (matrix) -- Multiplies every entry of a matrix by a scalar.
m (matrix) -- s (number) -- scale factorSolves the linear system m*x = b for x, exactly, across 2x2/3x3/general shapes. Returns explanatory text ('no unique solution') when the system has none.
m (matrix) -- b (number[]) -- the right-hand-side array, matching m's row countReduced row echelon form of any matrix (2x2, 3x3 or general rectangular), by exact Gauss-Jordan elimination -- fractions stay fractions.
m (matrix) -- The rank of a matrix: the number of pivots, exact. (rank() is a different thing: a tensor's number of indices, which is 2 for any matrix.) A decimal known only approximately is refused.
m (matrix) -- A basis of the null space, the solutions of A v = 0, as a list of vectors, one for each free column. An empty list when only v = 0 solves it.
m (matrix) -- A basis of the column space: the pivot columns of the matrix itself, as a list of vectors.
m (matrix) -- A basis of the row space: the nonzero rows of the reduced row echelon form.
m (matrix) -- The characteristic polynomial det(xI - A), monic and exact. Books that write det(A - xI) differ by (-1)^n; the roots are the same.
m (matrix) -- a square matrixThe distinct eigenvalues of a matrix of whole numbers and fractions: real ones smallest first, or all as complex numbers if any is not real. Exact where the characteristic polynomial's roots can be written exactly; otherwise decimals from a numerical solver.
m (matrix) -- a square matrixEach real eigenvalue with its multiplicity and a basis of its eigenspace (fields value, multiplicity, vectors). Refused if an eigenvalue is not real or is known only as a decimal.
m (matrix) -- a square matrixP and D with A = P D P^-1: the eigenvectors as P's columns, the eigenvalues on D's diagonal in the same order (fields P, D). Refused when there are too few eigenvectors.
m (matrix) -- a square matrixOrthogonal vectors spanning the columns of the matrix, by Gram-Schmidt, not scaled to length 1 so that they stay fractions. A column that depends on the earlier ones gives none.
m (matrix) -- its columns are the vectorsgramSchmidt's vectors scaled to length 1, exact (with square roots).
m (matrix) -- its columns are the vectorsThe projection of the vector b onto the column space of the matrix.
b (array) -- m (matrix) -- The x that brings A x closest to b, solving A^T A x = A^T b, exact. Refused when A's columns are dependent, as there is then no one answer.
m (matrix) -- b (array) -- P, L, U with P A = L U (fields P, L, U), exact, for a matrix of any shape: L unit lower triangular, a row swapped only where the pivot is zero.
m (matrix) -- Q and R with A = Q R (fields Q, R), by Gram-Schmidt, exact where the square roots allow: Q's columns orthonormal, R upper triangular with a positive diagonal. Refused when A's columns are dependent.
m (matrix) -- The singular value decomposition, NUMERIC (decimals, not exact), by one-sided Jacobi: A = U diag(S) V^T, the fields U, S, V, with S the nonzero singular values, largest first, and one column of U and V for each.
m (matrix) -- rref() as a step-by-step animation: one frame per elementary row operation (swap, scale, add a multiple of another row), each captioned with the exact operation, e.g. "R2 -> R2 - 2 R1".
m (matrix) -- How the matrices of this run (a page, a script) are written from here on: "square" puts them in square brackets, as the ACT prints them, "round" (the default) in round ones. Only the display changes, never a value.
style (text) -- "square" or "round"