LaTeX Equations with latexMatrix, Eqn and matrix2latex

The SVD

As a more complicated example, here we write out the LaTeX equation for the singular value decomposition (SVD) of a general \(n \times p\) matrix \(\mathbf{X}\) using Eqn() and latexMatrix(). In Rmd markup, Eqn() can be given an equation label (using the label argument), which will both label and number the equations.

Two calls to Eqn() produce separate equations in the output below. Both of these equations are numbered. (Eqn() uses the LaTeX equation environment, \begin{equation} ... \end{equation}, or equation* if the equation does not include a label). The two calls to Eqn() are rendered as separate equations, center aligned.

Eqn("\\mathbf{X} = \\mathbf{U} \\mathbf{\\Lambda} \\mathbf{V}^\\top", label='eq:svd')
Eqn("\\mathbf{X} =",
    latexMatrix("u", "n", "k"),
    latexMatrix("\\lambda", "k", "k", diag=TRUE),
    latexMatrix("v", "k", "p", transpose = TRUE), label='eq:svdmats')

This produces the two numbered equations:¹

\[\begin{equation} \tag{1} \mathbf{X} = \mathbf{U} \mathbf{\Lambda} \mathbf{V}^\top\end{equation}\]

\[\begin{equation} \tag{2} \begin{pmatrix} u_{11} & u_{12} & \cdots & u_{1k} \\ u_{21} & u_{22} & \cdots & u_{2k} \\ \vdots & \vdots & & \vdots \\ u_{n1} & u_{n2} & \cdots & u_{nk} \\ \end{pmatrix} \begin{pmatrix} \lambda_{1} & 0 & \cdots & 0 \\ 0 & \lambda_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_{k} \\ \end{pmatrix} \begin{pmatrix} v_{11} & v_{12} & \cdots & v_{1p} \\ v_{21} & v_{22} & \cdots & v_{2p} \\ \vdots & \vdots & & \vdots \\ v_{k1} & v_{k2} & \cdots & v_{kp} \\ \end{pmatrix}^\top \end{equation}\]

The matrix names in Equation (1) are printed in a boldface math font (\mathbf{}), typically used for matrices and vectors. Note that when using LaTeX code in R expressions each backslash (\) must be doubled (\\) in R because \ is the escape character.

Note that the first equation can be referenced because it was labeled: “As seen in Equation (1) ”. References to equations can entered in text using an inline call to ref(), e.g, `r ref("eq:svd")` (In Quarto, equation labels must be of the form #eq-label and equation references are of the form @eq-label)

Systems of equations

As another example, the chunk below shows a system of equations \(\mathbf{A} \mathbf{x} = \mathbf{b}\) written out using symbolic matrices.

Eqn(latexMatrix("a", nrow = "m", ncol="n", matrix="bmatrix"),
    latexMatrix("x", nrow = "n", ncol=1),
    Eqn_hspace(mid='='),
    latexMatrix("b", nrow = "m", ncol=1))

\[\begin{equation*} \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \\ \end{pmatrix} \quad=\quad\begin{pmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \\ \end{pmatrix} \end{equation*}\]

Extra symmetric white space is added via Eqn_hspace(), which can also be used for standard spacing such as \quad (default size), \,, '1cm' for \hspace{}, etc.

Section showEqn describes another way to display systems of equations.

aligned environment

You can also align separate equations relative to some symbol like an = sign to show separate steps of re-expression, using the option Eqn(..., align=TRUE). Alignment points are marked by & in LaTeX.

Show the singular value decomposition again, but now as two separate equations aligned after the = sign. Note the locations of the & operator for alignment, specified as the left-hand side (lhs) of the second equation.

Eqn("\\mathbf{X} & = \\mathbf{U} \\mathbf{\\Lambda} \\mathbf{V}^\\top",
    Eqn_newline(),
    ' & =',
    latexMatrix("u", "n", "k"),
    latexMatrix("\\lambda", "k", "k", diag=TRUE),
    latexMatrix("v", "k", "p", transpose = TRUE),
    align=TRUE)

\[\begin{align*} \mathbf{X} & = \mathbf{U} \mathbf{\Lambda} \mathbf{V}^\top \\ & =\begin{pmatrix} u_{11} & u_{12} & \cdots & u_{1k} \\ u_{21} & u_{22} & \cdots & u_{2k} \\ \vdots & \vdots & & \vdots \\ u_{n1} & u_{n2} & \cdots & u_{nk} \\ \end{pmatrix} \begin{pmatrix} \lambda_{1} & 0 & \cdots & 0 \\ 0 & \lambda_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_{k} \\ \end{pmatrix} \begin{pmatrix} v_{11} & v_{12} & \cdots & v_{1p} \\ v_{21} & v_{22} & \cdots & v_{2p} \\ \vdots & \vdots & & \vdots \\ v_{k1} & v_{k2} & \cdots & v_{kp} \\ \end{pmatrix}^\top \end{align*}\]

Note that in this example, there are three calls to latexMatrix(), wrapped inside Eqn(). Eqn_newline() emits a newline (\\) between equations.

Matrix products and transpose

The product of two matrices is given by %*% for "latexMatrix" objects. When the arguments are both numeric, the numeric result is evaluated and presented in LaTeX form.

A %*% B

But the result is symbolic if either argument is symbolic:

A %*% latexMatrix("b", 2, 2)

The LaTeX symbol for multiplication is a centered dot, \\cdot (\(\cdot\)), by default. This can be changed by changing options(latexMultSymbol), e.g, to use the \(\times\) symbol instead, use:

options(latexMultSymbol = "\\times")

The transpose, t() of "latexMatrix" objects is similarly straightforward. This directly transposes the matrix, as opposed to superscript notation, \(\mathbf{D}^\top\) or \(\mathbf{D}^\prime\) which is implicit.

D

## \begin{pmatrix} 
##   d_{11} & d_{12} & d_{13} \\ 
##   d_{21} & d_{22} & d_{23} \\ 
## \end{pmatrix}

t(D)

## \begin{pmatrix}  
## d_{11} & d_{21} \\ 
## d_{12} & d_{22} \\ 
## d_{13} & d_{23} \\ 
## \end{pmatrix}

M %*% t(D)

## \begin{pmatrix}  
## a \cdot d_{11} + d \cdot d_{12} + g \cdot d_{13} & a \cdot d_{21} + d \cdot d_{22} + g \cdot d_{23} \\ 
## b \cdot d_{11} + e \cdot d_{12} + h \cdot d_{13} & b \cdot d_{21} + e \cdot d_{22} + h \cdot d_{23} \\ 
## c \cdot d_{11} + f \cdot d_{12} + i \cdot d_{13} & c \cdot d_{21} + f \cdot d_{22} + i \cdot d_{23} \\ 
## \end{pmatrix}

The matrix product in the previous example typesets as

\[\begin{equation*} \begin{pmatrix} a \cdot d_{11} + d \cdot d_{12} + g \cdot d_{13} & a \cdot d_{21} + d \cdot d_{22} + g \cdot d_{23} \\ b \cdot d_{11} + e \cdot d_{12} + h \cdot d_{13} & b \cdot d_{21} + e \cdot d_{22} + h \cdot d_{23} \\ c \cdot d_{11} + f \cdot d_{12} + i \cdot d_{13} & c \cdot d_{21} + f \cdot d_{22} + i \cdot d_{23} \\ \end{pmatrix} \end{equation*}\]

Determinants & inverse

The determinant is computed recursively by finding the cofactors of the first row of the matrix, i.e., \(\det(\mathbf{A}_{n \times n}) = \Sigma_j^n \, a_{ij}\: C_{ij}\) where the cofactors \(C_{ij}\) involve the determinants of the \((n-1) \times (n-1)\) minors \(\mathbf{A}_{ij}\). (See the vignette Evaluation of determinants for explanation.)

The method is applicable to a matrix of any order, although beyond an order-3 matrix, the resulting expression gets very long. A couple of examples:

A

## \begin{pmatrix} 
##  1 &  0 \\ 
## -3 &  1 \\ 
## \end{pmatrix}

determinant(A)

## [1] "1 \\cdot 1 - 0 \\cdot (-3)"

M

## \begin{pmatrix} 
## a & d & g \\ 
## b & e & h \\ 
## c & f & i \\ 
## \end{pmatrix}

determinant(M)

## [1] "a \\cdot (e \\cdot i - h \\cdot f) - d \\cdot (b \\cdot i - h \\cdot c) + g \\cdot (b \\cdot f - e \\cdot c)"

The determinant of a matrix is a single number. determinant() returns the expression that computes it in LaTeX notation, using \\cdot to represent \(\cdot\) for multiplication.

Typesetting the output from the last command produces

\[\begin{equation*} a \cdot (e \cdot i - h \cdot f) - d \cdot (b \cdot i - h \cdot c) + g \cdot (b \cdot f - e \cdot c)\end{equation*}\]

The inverse of a square matrix is computed from its determinant and adjoint matrix; for example:

solve(A)

## \begin{pmatrix}  
## 1 & 0 \\ 
## 3 & 1 \\ 
## \end{pmatrix}

Specifying the argument simplify = TRUE to solve() puts the inverse determinant before the adjoint matrix and returns a latex expression rather than a "latexMatrix" object; for example:

solve(M, simplify=TRUE)

## [1] "\\frac{1}{a \\cdot (e \\cdot i - h \\cdot f) - d \\cdot (b \\cdot i - h \\cdot c) + g \\cdot (b \\cdot f - e \\cdot c)} \n\\begin{pmatrix}  \ne \\cdot i - h \\cdot f    & -(d \\cdot i - g \\cdot f) & d \\cdot h - g \\cdot e    \\\\ \n-(b \\cdot i - h \\cdot c) & a \\cdot i - g \\cdot c    & -(a \\cdot h - g \\cdot b) \\\\ \nb \\cdot f - e \\cdot c    & -(a \\cdot f - d \\cdot c) & a \\cdot e - d \\cdot b    \\\\ \n\\end{pmatrix}\n"

which typesets as

\[\begin{equation*} \frac{1}{a \cdot (e \cdot i - h \cdot f) - d \cdot (b \cdot i - h \cdot c) + g \cdot (b \cdot f - e \cdot c)} \begin{pmatrix} e \cdot i - h \cdot f & -(d \cdot i - g \cdot f) & d \cdot h - g \cdot e \\ -(b \cdot i - h \cdot c) & a \cdot i - g \cdot c & -(a \cdot h - g \cdot b) \\ b \cdot f - e \cdot c & -(a \cdot f - d \cdot c) & a \cdot e - d \cdot b \\ \end{pmatrix} \end{equation*}\] We can also supply values for the elements of the matrix to obtain a numeric inverse:

MASS::fractions(as.double(solve(M), 
                          locals=c(a=1, b=0, c=-2, d=-4, e=7, f=-1, g=0, h=4, i=6)))

##      [,1]  [,2]  [,3] 
## [1,] 23/39  4/13 -8/39
## [2,] -4/39  1/13 -2/39
## [3,]  7/39  3/26  7/78

MASS::fractions(det(as.double(M, 
                              locals=c(a=1, b=0, c=-2, d=-4, e=7, f=-1, g=0, h=4, i=6))))

## [1] 78

Linear hypotheses

As an example of the more general use of these functions, consider the general linear hypothesis used to test hypotheses and contrasts in linear models. We consider a multivariate regression model \(\mathbf{Y} = \mathbf{X} \mathbf{B} + \mathbf{E}\) with \(q\) regressors \(\mathbf{x}_0, \mathbf{x}_1, \dots, \mathbf{x}_q\) (including the constant \(\mathbf{x}_0\) for the intercept) and \(p\) responses, \(\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_p\).

Suppose we want to test the hypothesis that a subset of rows (predictors) and/or columns (responses) simultaneously have null effects. This can be expressed in the general linear test, \[ \mathcal{H}_0 : \mathbf{C}_{h \times q} \, \mathbf{B}_{q \times p} = \mathbf{0}_{h \times p} \: , \] where \(\mathbf{C}\) is a full rank \(h \le q\) hypothesis matrix of constants, that selects subsets or linear combinations (contrasts) of the coefficients in \(\mathbf{B}\) to be tested in a \(h\) degree-of-freedom hypothesis.

For example, for a multivariate regression model with three responses \(y_1, y_2, y_3\) and three predictors \(x_1, x_2, x_3\), the coefficients \(\mathbf{B}\) are given by the following latexMatrix() expression, where several arguments are used to: (a) start row indices at zero (zero.based); (b) make the column indices a subscript of \(y\) (prefix.col); (c) insert a comma between row/column subscripts.

(B <- latexMatrix('\\beta', ncol = 3, nrow=4, 
                 comma=TRUE, prefix.col = 'y_',
                 zero.based=c(TRUE, FALSE)))

We can test the hypothesis that neither \(x_2\) nor \(x_3\) contribute at all to the predicting the \(y\)s in terms of the hypothesis that the coefficients for the corresponding rows of \(\mathbf{B}\) are zero. To do this, we specify a 2-row \(\mathbf{C}\) matrix that simply selects those rows:

(C <- latexMatrix(matrix(c(0, 1, 0, 0,
                           0, 0, 1, 0), nrow=2, byrow=TRUE), 
                 matrix = "bmatrix"))

Then, the hypothesis to be tested can be expressed as follows, using Eqn() to wrap a set of LaTeX expressions and calls to matlib functions.

B0 <- latexMatrix('\\beta', ncol = 3, nrow=2, comma=TRUE, prefix.col = 'y_')
Eqn("\\mathcal{H}_0 : \\mathbf{C} \\mathbf{B} & = ",
    C, B,
    Eqn_newline(), 
    '& =',
    B0,
    "= \\mathbf{0}_{(2 \\times 3)}", 
    align=TRUE)

\[\begin{align*} \mathcal{H}_0 : \mathbf{C} \mathbf{B} & = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} \begin{pmatrix} \beta_{0,y_{1}} & \beta_{0,y_{2}} & \beta_{0,y_{3}} \\ \beta_{1,y_{1}} & \beta_{1,y_{2}} & \beta_{1,y_{3}} \\ \beta_{2,y_{1}} & \beta_{2,y_{2}} & \beta_{2,y_{3}} \\ \beta_{3,y_{1}} & \beta_{3,y_{2}} & \beta_{3,y_{3}} \\ \end{pmatrix} \\ & =\begin{pmatrix} \beta_{1,y_{1}} & \beta_{1,y_{2}} & \beta_{1,y_{3}} \\ \beta_{2,y_{1}} & \beta_{2,y_{2}} & \beta_{2,y_{3}} \\ \end{pmatrix} = \mathbf{0}_{(2 \times 3)}\end{align*}\]

In this example, note that the R objects C, B and B0 are the results of latexMatrix() calls, which are character strings containing LaTeX expressions.

Partitioned matrices, indexing & binding

Matrix notation sometimes portrays matrices whose elements are themselves matrices and vectors (rather than scalars) in order to show a higher-level structure. Such matrices, called partitioned or block matrices have similar arithmetic and algebraic properties to those of ordinary matrices.

For example, the code below represents a \(4 \times 4\) matrix \(\mathbf{M}\), which is partitioned in \(2 \times 2\) blocks, which are labeled \(\mathbf{M}_{i,j}\).

M <- latexMatrix("m", 4, 4)
Mpart <- latexMatrix('\\mathbf{M}', nrow = 2, ncol = 2, comma = TRUE)
Eqn("\\mathbf{M} =", Mpart,
    " =", M)

## 
## \begin{equation*}
## \mathbf{M} =\begin{pmatrix} 
##   \mathbf{M}_{1,1} & \mathbf{M}_{1,2} \\ 
##   \mathbf{M}_{2,1} & \mathbf{M}_{2,2} \\ 
## \end{pmatrix}
##  =\begin{pmatrix} 
##   m_{11} & m_{12} & m_{13} & m_{14} \\ 
##   m_{21} & m_{22} & m_{23} & m_{24} \\ 
##   m_{31} & m_{32} & m_{33} & m_{34} \\ 
##   m_{41} & m_{42} & m_{43} & m_{44} \\ 
## \end{pmatrix}
## \end{equation*}

This typesets as:

\[\begin{equation*} \mathbf{M} =\begin{pmatrix} \mathbf{M}_{1,1} & \mathbf{M}_{1,2} \\ \mathbf{M}_{2,1} & \mathbf{M}_{2,2} \\ \end{pmatrix} =\begin{pmatrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \\ m_{41} & m_{42} & m_{43} & m_{44} \\ \end{pmatrix} \end{equation*}\]

Just as rows and columns can be selected using X[rows, cols] indexing for ordinary matrices, the same operator can be used for LaTeX matrices, e.g., M[rows, cols]. The following extracts 4 the sub-matrices of M:

M11 <- M[1:2, 1:2] |> print()

## \begin{pmatrix}  
## m_{11} & m_{12} \\ 
## m_{21} & m_{22} \\ 
## \end{pmatrix}

M12 <- M[1:2, 3:4]
M21 <- M[3:4, 1:2]
M22 <- M[3:4, 3:4]

The operations of joining matrices by rows, with rbind(), and by columns, with cbind() are also defined for "latexMatrices". This code puts the 4 pieces of \(\mathbf{M}\) back together:

rbind(
  cbind(M11, M12),
  cbind(M21, M22)
)

## \begin{pmatrix}  
## m_{11} & m_{12} & m_{13} & m_{14} \\ 
## m_{21} & m_{22} & m_{23} & m_{24} \\ 
## m_{31} & m_{32} & m_{33} & m_{34} \\ 
## m_{41} & m_{42} & m_{43} & m_{44} \\ 
## \end{pmatrix}

And, of course you can also format the sub-matrices together using Eqn():

Eqn(M11, M12,
    Eqn_newline(),
    M21, M22,
    align = TRUE)

\[\begin{align*} \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \\ \end{pmatrix} \begin{pmatrix} m_{13} & m_{14} \\ m_{23} & m_{24} \\ \end{pmatrix} \\ \begin{pmatrix} m_{31} & m_{32} \\ m_{41} & m_{42} \\ \end{pmatrix} \begin{pmatrix} m_{33} & m_{34} \\ m_{43} & m_{44} \\ \end{pmatrix} \end{align*}\]

Finally, the partition() function alters the print representation of a matrix using horizontal and vertical lines separating the sub-matrices. It does this by re-wrapping the matrix in a LaTeX \begin{array} ... \end{array} environment, using | in {c c | c c} for the vertical lines and \hline for horizontal lines. This may be the simplest way to portray partitioned matrices in writing.

partition(M, rows=2, columns=2)

Note that partition() can show more than one horizontal and vertical partition lines (or no line at all):

partition(M, rows=c(1,3), columns=c(1,3))

Using this notation, we can illustrate matrix arithmetic with partitioned matrices. For example, assuming the partitions of matrices \(\mathbf{C}\) and \(\mathbf{D}\) are of the same size, their sum is just the sum of corresponding sub-matrices:

C <- latexMatrix("\\mathbf{C}", 2, 2)
D <- latexMatrix("\\mathbf{D}", 2, 2)
Eqn("\\mathbf{C} + \\mathbf{D} =",
    C, "+", D, "=", 
    C + D)

\[\begin{equation*} \mathbf{C} + \mathbf{D} =\begin{pmatrix} \mathbf{C}_{11} & \mathbf{C}_{12} \\ \mathbf{C}_{21} & \mathbf{C}_{22} \\ \end{pmatrix} +\begin{pmatrix} \mathbf{D}_{11} & \mathbf{D}_{12} \\ \mathbf{D}_{21} & \mathbf{D}_{22} \\ \end{pmatrix} =\begin{pmatrix} \mathbf{C}_{11} + \mathbf{D}_{11} & \mathbf{C}_{12} + \mathbf{D}_{12} \\ \mathbf{C}_{21} + \mathbf{D}_{21} & \mathbf{C}_{22} + \mathbf{D}_{22} \\ \end{pmatrix} \end{equation*}\]

Kronecker products

The Kronecker product of two matrices, \(\mathbf{A}_{m \times n} \otimes \mathbf{B}_{p \times q}\) is the \(mp \times nq\) block matrix consisting of each element \(a_{ij}\) multiplied by \(\mathbf{B}\). This has many uses in statistics, among these the nice result (Bock 1975; Sunwoo 1996) that the design matrix \(\mathbf{X}\) in the linear ANOVA model for factors A, B, C, … can be generated as the Kronecker product of their contrast matrices \(\mathbf{C}_A, \mathbf{C}_B, \mathbf{C}_C \dots\), each preceded by the unit vector \(\mathbf{1}\).

\[ \mathbf{X}_{ABC\dots} = [\mathbf{1} \mid \mathbf{C}_A] \;\otimes\; [\mathbf{1} \mid \mathbf{C}_B] \;\otimes\; [\mathbf{1} \mid \mathbf{C}_B] \;\otimes\; \dots \]

This is implemented in the %O% operator and the kronecker() function in the package. For example,

A <- matrix(1:4, nrow = 2) |> 
  latexMatrix() |> print()

B <- matrix(5:8, nrow = 2) |> 
  latexMatrix() |> print()

kronecker(A, B) |> partition(rows = 2, columns = 2)

You can also use Eqn() to illustrate the definition of the Kronecker product more explicitly. In the following, KAB is the product in symbolic form; as.double() is used to evaluate the result numerically.

Bmat <- latexMatrix('\\mathbf{B}', ncol=1, nrow=1)
KABmat <- kronecker(A, Bmat)
KAB <- kronecker(A, B)

Eqn("\\mathbf{A} \\otimes \\mathbf{B} = &",
    KABmat,
    Eqn_newline(), Eqn_vspace("1.5ex"), "= & ",
    KAB |> partition(rows = 2, columns = 2),
    Eqn_newline(), Eqn_vspace("1.5ex"), "= & ",
    latexMatrix(as.double(KAB)) |> partition(rows = 2, columns = 2),
    align = TRUE)

\[\begin{align*} \mathbf{A} \otimes \mathbf{B} = &\begin{pmatrix} 1 \cdot \mathbf{B} & 3 \cdot \mathbf{B} \\ 2 \cdot \mathbf{B} & 4 \cdot \mathbf{B} \\ \end{pmatrix} \\ \vspace{1.5ex} = & \begin{pmatrix} \begin{array}{c c | c c} 1 \cdot 5 & 1 \cdot 7 & 3 \cdot 5 & 3 \cdot 7\\ 1 \cdot 6 & 1 \cdot 8 & 3 \cdot 6 & 3 \cdot 8\\ \hline 2 \cdot 5 & 2 \cdot 7 & 4 \cdot 5 & 4 \cdot 7\\ 2 \cdot 6 & 2 \cdot 8 & 4 \cdot 6 & 4 \cdot 8\\ \end{array} \end{pmatrix} \\ \vspace{1.5ex} = & \begin{pmatrix} \begin{array}{c c | c c} 5 & 7 & 15 & 21\\ 6 & 8 & 18 & 24\\ \hline 10 & 14 & 20 & 28\\ 12 & 16 & 24 & 32\\ \end{array} \end{pmatrix}\end{align*}\]