Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics, http://friendly.github.io/matlib/

Version 0.9.9

These functions were originally designed 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, e.g., R() for the rank of a matrix, or tr() for matrix trace.

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, e.g., Det() for a matrix determinant, Inverse() for a matrix inverse, using gaussianElimination() to show the steps.

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. For example,

• showEqn(A, b) shows the matrix equations $A x = b$ in text or LaTeX form, while plotEqn(A, b) and plotEqn3d(A, b) plots those equations in 2D or 3D space.

• matrix2latex() and latexMatrix() facilitate writing matrix equations in LaTeX

• vectors(), vectors3d() plot geometric vector diagrams in 2D or 3D, with other functions to draw angles and arcs.

• regvec3d() calculates and plot vectors representing a bivariate regression model, lm(y ~ x1 + x2)

## Installation

Get the released version from CRAN:

 install.packages("matlib")

Or the development version from my R-universe:

install.packages('matlib', repos = c('https://friendly.r-universe.dev', 'https://cloud.r-project.org'))

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

 if (!require(remotes)) install.packages("remotes")
remotes::install_github("friendly/matlib", build_vignettes = TRUE)

The functions that draw 3D graphs use the rgl package. On macOS, rgl requires that XQuartz be installed. After installing XQuartz, itβs necessary either to log out of and back into your macOS account or to reboot your Mac.

## Topics

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 matrix X
• mpower(A, p) - matrix powers for a square symmetric matrix
• xprod(...) - vector cross-product
1. Determinants: functions for calculating determinants by cofactor expansion
• minor() - Minor of A[i,j]
• cofactor() - Cofactor of A[i,j]
• rowMinors() - Row minors of A[i,]
• rowCofactors() - Row cofactors of A[i,]
• Det() - Determinants by elimination or eigenvalues
1. Elementary row operations: functions for solving linear equations βmanuallyβ 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 $\mathbf{A x = b}$
• showEqn(A, b) - show matrices (A, b) as linear equations
• plotEqn(A, b), plotEqn3d(A, b) - plot matrices (A, b) as linear equations
1. Gaussian elimination: functions for illustrating Gaussian elimination for solving systems of linear equations of the form $\mathbf{A x = b}$. 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, $\mathbf{X}^{-1}$
• 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(X) - 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, $mathbf{X = U D V}$
• 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
• arrows3d() - draw nice 3D arrows
• corner(), arc() - draw a corner or arc showing the angle between two vectors in 2D/3D
• pointOnLine() - position of a point along a line
• vectors(), vectors3d() - plot geometric vector diagrams in 2D/3D
• regvec3d() - calculate and plot vectors representing a bivariate regression model, lm(y ~ x1 + x2) in mean-deviation form.
1. Matrix equations
• matrix2latex(): Convert matrix to LaTeX equation
• latexMatrix(): Create and manipulate LaTeX repesentations of matrices

### 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