Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics,

Version 0.9.4

These functions are mainly for tutorial purposes in teaching & learning matrix algebra ideas using R. In some cases, functions are provided for concepts or computations available elsewhere in R, but where the name is not obvious. In other cases, we provide cover functions to show or demonstrate an algorithm in more detail, sometimes providing a verbose = argument to print the details of computations.

In addition, a collection of functions are provided for drawing vector diagrams in 2D and 3D, illustrating various concepts of linear algebra more concretely than has been available before.


Get the released version from CRAN:

The development version can be installed to your R library directly from this repo via:

This installs the package from the source and creates the package vignettes, so you will need to have R Tools installed on your system. R Tools for Windows takes you to the download page for Windows. R Tools for Mac OS X has the required programs for Mac OS X.


The functions in this package are grouped under the following topics

  1. Convenience functions:
  • tr() - trace of a matrix
  • R() - rank of a matrix
  • J() - constant vector, matrix or array
  • len() - Euclidean length of a vector or columns of a matrix
  • vec() - vectorize a matrix
  • Proj(y, X) - projection of vector y on columns of X
  • mpower(A, p) - matrix powers for a square symmetric matrix
  • xprod(...) - vector cross-product
  1. Determinants: functions for calculating determinants by cofactor expansion
  1. Elementary row operations: functions for solving linear equations “manually” by the steps used in row echelon form and Gaussian elimination
  • rowadd() - Add multiples of rows to other rows
  • rowmult() - Multiply rows by constants
  • rowswap() - Interchange two rows of a matrix
  1. Linear equations: functions to illustrate linear equations of the form Ax=b
  1. Gaussian elimination: functions for illustrating Gaussian elimination for solving systems of linear equations of the form Ax=b. These functions provide a verbose=TRUE argument to show the intermediate steps and a fractions=TRUE argument to show results using MASS::fractions.
  • gaussianElimination(A, B) - reduces (A, B) to (I, A^{-1} B)
  • Inverse(X), inv() - uses gaussianElimination to find the inverse of X
  • echelon(X) - uses gaussianElimination to find the reduced echelon form of X
  • Ginv(X) - uses gaussianElimination to find the generalized inverse of X
  • LU(X) - LU decomposition of a matrix X
  • cholesky() - calculates a Cholesky square root of a matrix
  • swp() - matrix sweep operator
  1. Eigenvalues: functions to illustrate the algorithms for calculating eigenvalues and eigenvectors and related matrix decompositions and generalizations.
  • Eigen() - eigenvalues and eigenvectors
  • SVD() - singular value decomposition
  • powerMethod() - find the dominant eigenvector using the power method
  • showEig() - draw eigenvectors on a 2D scatterplot with a dataEllipse
  • MoorePenrose() - illustrates how the Moore-Penrose inverse can be calculated using SVD()
  1. Vector diagrams: functions for drawing vector diagrams in 2D and 3D

Vignettes and presentations

A small collection of vignettes is now available. Use browseVignettes("matlib") to explore them.

Vignette Title
det-ex1 Properties of determinants
det-ex2 Evaluation of determinants
inv-ex1 Inverse of a matrix
inv-ex2 Matrix inversion by elementary row operations
ginv Generalized inverse
eigen-ex1 Eigenvalues and Eigenvectors: Properties
eigen-ex2 Eigenvalues: Spectral Decomposition
linear-equations Solving Linear Equations
gramreg Gram-Schmidt Orthogonalization and Regression
data-beta Vector Spaces of Least Squares and Linear Equations

See also: Fox & Friendly, Visualizing Simultaneous Linear Equations, Geometric Vectors, and Least-Squares Regression with the matlib Package for R, June 2016, useR! Conference, Stanford.