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Linear Algebra

Using

using DoubleFloats, LinearAlgebra

Vectors and Matrices

using DoubleFloats, LinearAlgebra

n = 25
vector = rand(Double64, n)
matrix = reshape(rand(Double64,n*n),n,n)

Matrix Predicates

  • iszero, isone, isdiag
  • issquare, issymmetric, ishermitian
  • isposdef, isposdef!
  • istril, istriu

Matrix Operations

  • +, -, *, /
  • rank, cond, norm, opnorm
  • diagind, diag
  • transpose, adjoint
  • inv, pinv

Matrix Functions

  • det, logdet, logabsdet, tr
  • eigvals, eigvals!, eigvecs, eigen
  • svdvals, svdvals!, svd

Matrix Factorizations

  • general: lu, lu!, qr, qr!, lq
  • square: schur, schur!, hessenberg, hessenberg!
  • square+symmetric, Hermitian: cholesky, cholesky!, bunchkaufman

eigen for DoubleFloat matrices follows the LAPACK-route conventions: symmetric/Hermitian input yields real eigenvalues and orthonormal vectors, a real matrix with an all-real spectrum yields a real factorization, and eigenvalues are sorted by (real, imag).

Matrix Equations

  • sylvester(A, B, C) – solves AX + XB + C = 0
  • lyap(A, C) – solves AX + XA' + C = 0

Functions of Matrices (any square matrix)

These use dense Schur-based and scaling-and-squaring algorithms, so they are correct for defective (non-diagonalizable) matrices too, and real matrices give real results whenever the mathematical result is real:

  • exp, expm1, log, log1p, log2, log10, sqrt
  • matrix^power
  • sin, cos, sincos, tan, csc, sec, cot
  • sinh, cosh, tanh, csch, sech, coth

Functions of Matrices (diagonalizable & square only)

These use eigen-diagonalization via matrixfunction(function, matrix):

  • cbrt, sinpi, cospi
  • asin, acos, atan, acsc, asec, acot
  • asinh, acosh, atanh, acsch, asech, acoth