Introduction

This vignette is based on caracas version r packageVersion("caracas"). caracas is avavailable on CRAN at [https://cran.r-project.org/package=caracas] and on github at [https://github.com/r-cas/caracas].

Elementary matrix operations

Creating matrices / vectors

We now show different ways to create a symbolic matrix:

A <- matrix(c("a", "b", "0", "1"), 2, 2) |> as_sym()
A
A <- matrix_(c("a", "b", "0", "1"), 2, 2) # note the '_' postfix
A
A <- as_sym("[[a, 0], [b, 1]]")
A

A2 <- matrix_(c("a", "b", "c", "1"), 2, 2)
A2

B <- matrix_(c("a", "b", "0", 
              "c", "c", "a"), 2, 3)
B

b <- matrix_(c("b1", "b2"), nrow = 2)

D <- diag_(c("a", "b")) # note the '_' postfix
D

Note that a vector is a matrix in which one of the dimensions is one.

Matrix-matrix sum and product

A + A2
A %*% B

Hadamard (elementwise) product

A * A2

Vector operations

x <- as_sym(paste0("x", 1:3))
x
x + x
1 / x
x / x

Reciprocal matrix

reciprocal_matrix(A2)
reciprocal_matrix(A2, num = "2*a")

Matrix inverse; solve system of linear equations

Solve \(Ax=b\) for \(x\):

inv(A)
x <- solve_lin(A, b)
x
A %*% x ## Sanity check

Generalized (Penrose-Moore) inverse; solve system of linear equations [TBW]

M <- as_sym("[[1, 2, 3], [4, 5, 6]]")
pinv(M)
B <- as_sym("[[7], [8]]") 
B
z <- do_la(M, "pinv_solve", B)
print(z, rowvec = FALSE) # Do not print column vectors as transposed row vectors

More special linear algebra functionality

Below we present convenient functions for performing linear algebra operations. If you need a function in SymPy for which we have not supplied a convinience function (see https://docs.sympy.org/latest/modules/matrices/matrices.html), you can still call it with the do_la() (short for “do linear algebra”) function presented at the end of this section.

QR decomposition

A <- matrix(c("a", "0", "0", "1"), 2, 2) |> as_sym()
A
qr_res <- QRdecomposition(A)
qr_res$Q
qr_res$R

Eigenvalues and eigenvectors

eigenval(A)

evec <- eigenvec(A)
evec
evec1 <- evec[[1]]$eigvec
evec1
simplify(evec1)

lapply(evec, function(l) simplify(l$eigvec))

Inverse, Penrose-Moore pseudo inverse

inv(A)
pinv(cbind(A, A)) # pseudo inverse

Additional functionality for linear algebra

do_la short for “do linear algebra”

args(do_la)

The above functions can be called:

do_la(A, "QRdecomposition") # == QRdecomposition(A)
do_la(A, "inv")             # == inv(A)
do_la(A, "eigenvec")        # == eigenvec(A)
do_la(A, "eigenvals")       # == eigenval(A)

Characteristic polynomial

cp <- do_la(A, "charpoly")
cp
as_expr(cp)

Rank

do_la(A, "rank")

Cofactor

A <- matrix(c("a", "b", "0", "1"), 2, 2) |> as_sym()
A
do_la(A, "cofactor", 0, 1)
do_la(A, "cofactor_matrix")

Echelon form

do_la(cbind(A, A), "echelon_form")

Cholesky factorisation

B <- as_sym("[[9, 3*I], [-3*I, 5]]")
B
do_la(B, "cholesky")

Gram Schmidt

B <- t(as_sym("[[ 2, 3, 5 ], [3, 6, 2], [8, 3, 6]]"))
do_la(B, "GramSchmidt")

Reduced row-echelon form (rref)

B <- t(as_sym("[[ 2, 3, 5 ], [4, 6, 10], [8, 3, 6] ]"))
B
B_rref <- do_la(B, "rref")
B_rref

Column space, row space and null space

B <- matrix(c(1, 3, 0, -2, -6, 0, 3, 9, 6), nrow = 3) |> as_sym()
B
columnspace(B)
rowspace(B)
x <- nullspace(B)
x
rref(B)
B %*% x

Singular values, svd

B <- t(as_sym("[[ 2, 3, 5 ], [4, 6, 10], [8, 3, 6], [8, 3, 6] ]"))
B
singular_values(B)

11 - Linear algebra in caracas