\chapter{Matrices}
\label{chap:matrices}

Matrices are one- or two-dimensional arrays of double-precision
floating-point numbers. Matrices have rows and columns, and that's it
(but see Section~\ref{sec:arrays} for means of constructing
higher-dimensional compound objects).

\section{Matrix indexing}
\label{sec:mat-index}

Individual matrix elements are accessed through the \verb|[r,c]|
syntax, where indexing starts at 1. For example, \texttt{X[3,4]}
indicates the element of $X$ on the third row, fourth column. For
example,
\begin{code}
  matrix X = zeros(2,3)
  X[2,1] = 4
  print X
\end{code}
produces
\begin{code}
X (2 x 3)

  0   0   0 
  4   0   0 
\end{code}

Here are some more advanced ways to access matrix elements:
\begin{enumerate}
\item In case the matrix has only one row (column), the column (row)
  specification can be omitted, as in \texttt{x[3]}.
\item Including the comma but omitting the row or column specification
  means ``take them all'', as in \texttt{x[4,]} (fourth row, all columns).
\item For square matrices, the special syntax \texttt{x[diag]} can be
  used to access the diagonal.
\item Consecutive rows or columns can be specified via the colon
  (\texttt{:}) character, as in \texttt{x[,2:4]} (columns 2 to 4).
  But note that, unlike some other matrix languages, the syntax
  \texttt{[m:n]} is illegal if $m>n$.
\item It is possible to use a vector to hold indices to a matrix. E.g.\
  if $e = [2,3,6]$, then \texttt{X[,e]} contains the second, third and
  sixth columns of $X$.
\end{enumerate}
Moreover, matrices can be empty (zero rows and columns). 

In the example above, the matrix \texttt{X} was constructed using
the function \texttt{zeros()}, whose meaning should be obvious, but
matrix elements can also be specified directly, as in
\begin{code}
scalar a = 2*3
matrix A = { 1, 2, 3 ; 4, 5, a }
\end{code}
The matrix is defined by rows; the elements on each row are separated
by commas and rows are separated by semicolons.  The whole expression
must be wrapped in braces.  Spaces within the braces are not
significant. The above expression defines a $2\times3$ matrix.

Note that each element should be a numerical value, the name of a
scalar variable, or an expression that evaluates to a scalar. In the
example above the scalar \texttt{a} was first assigned a value and
then used in matrix construction. (Also note, in passing, that
\texttt{a} and \texttt{A} are two separate identifiers, due to
case-sensitivity.)

\section{Matrix operations}
\label{sec:mat-op}

Matrix sum, difference and product are obtained via \texttt{+},
\texttt{-} and \texttt{*}, respectively. The prime operator
(\texttt{'}) can act as a unary operator, in which case it transposes
the preceding matrix, or as a binary operator, in which case it acts
as in ordinary matrix algebra, multiplying the transpose of the first
matrix into the second one.\footnote{In fact, in this case an
  optimized algorithm is used; you should always use \texttt{a'a}
  instead of \texttt{a'*a} for maximal precision and performance.}
Errors are flagged if conformability is a problem. For example:
\begin{code}
  matrix a = {11, 22 ; 33, 44}  # a is square 2 x 2
  matrix b = {1,2,3; 3,2,1}     # b is 2 x 3

  matrix c = a'         # c is the transpose of a
  matrix d = a*b        # d is a 2x3 matrix equal to a times b

  matrix gina = b'd     # valid: gina is 3x3
  matrix lina = d + b   # valid: lina is 2x3

  /* -- these would generate errors if uncommented ----- */

  # pina = a + b  # sum non-conformability
  # rina = d * b  # product non-conformability
\end{code}

Other noteworthy matrix operators include \texttt{\^} (matrix power),
\texttt{**} (Kronecker product), and the ``concatenation'' operators,
\verb|~| (horizontal) and \texttt{|} (vertical). Readers are invited
to try them out by running the following code
\begin{code}
matrix A = {2,1;0,1}
matrix B = {1,1;1,0}

matrix KP = A ** B
matrix PWR = A^3 
matrix HC = A ~ B
matrix VC = A | B

print A B KP PWR HC VC
\end{code}
Note, in particular, that $A^3 = A \cdot A \cdot A$, which is different
from what you get by computing the cubes of each element of $A$
separately.

Hansl also supports matrix left- and right-``division'', via the
\verb'\' and \verb'/' operators, respectively. The expression
\verb|A\b| solves $Ax = b$ for the unknown $x$. $A$ is assumed to be
an $m \times n$ matrix with full column rank. If $A$ is square the
method is LU decomposition. If $m > n$ the QR decomposition is used to
find the least squares solution. In most cases, this is numerically
more robust and more efficient than inverting $A$ explicitly.

Element-by-element operations are supported by the so-called ``dot''
operators, which are obtained by putting a dot (``\texttt{.}'') before
the corresponding operator. For example, the code
\begin{code}
A = {1,2; 3,4}
B = {-1,0; 1,-1}
eval A * B
eval A .* B
\end{code}
produces
\begin{code}
   1   -2 
   1   -4 

  -1    0 
   3   -4 
\end{code}

It's easy to verify that the first operation performed is regular
matrix multiplication $A \cdot B$, whereas the second one is the
Hadamard (element-by-element) product $A \odot B$. In fact, dot
operators are more general and powerful than shown in the example
above; see the chapter on matrices in \GUG{} for details.

Dot and concatenation operators are less rigid than ordinary matrix
operations in terms of conformability requirements: in most cases
hansl will try to do ``the obvious thing''. For example, a common
idiom in hansl is \texttt{Y = X ./ w}, where $X$ is an $n \times k$
matrix and $w$ is an $n \times 1$ vector. The result $Y$ is an $n
\times k$ matrix in which each row of $X$ is divided by the
corresponding element of $w$. In proper matrix notation, this
operation should be written as
\[
  Y = \langle w \rangle^{-1} X,
\]
where the $\langle \cdot \rangle$ indicates a diagonal
matrix. Translating literally the above expression would imply
creating a diagonal matrix out of $w$ and then inverting it, which is
computationally much more expensive than using the dot operation. A
detailed discussion is provided in \GUG.

Hansl provides a reasonably comprehensive set of matrix functions,
that is, functions that produce and/or operate on matrices. For a
full list, see the \GCR, but a basic ``survival kit'' is provided
in Table~\ref{tab:essential-matfuncs}.  Moreover, most scalar
functions, such as \texttt{abs(), log()} etc., will operate on a
matrix element-by-element.

\begin{table}[htbp]
  \centering
  \small
  \begin{tabular}{rp{0.6\textwidth}}
    \textbf{Function(s)} & \textbf{Purpose} \\
    \hline
    \texttt{rows(X), cols(X)} & return the number of rows and columns
    of $X$, respectively \\
    \texttt{zeros(r,c), ones(r,c)} & produce matrices with $r$ rows
    and $c$ columns, filled with zeros and ones, respectively \\
    \texttt{mshape(X,r,c)} & rearrange the elements of $X$ into a
    matrix with $r$ rows and $c$ columns \\
    \texttt{I(n)} & identity matrix of size $n$ \\
    \texttt{seq(a,b)} & generate a row vector containing integers from
    $a$ to $b$ \\
    \texttt{inv(A)} & invert, if possible, the matrix $A$ \\
    \texttt{maxc(A), minc(A), meanc(A)} & return a row vector
    with the max, min, means of each column of $A$, respectively\\
    \texttt{maxr(A), minr(A), meanr(A)} & return a column vector
    with the max, min, means of each row of $A$, respectively\\
    \texttt{mnormal(r,c), muniform(r,c)} & generate $r \times c$
    matrices filled with standard Gaussian and uniform pseudo-random
    numbers, respectively \\
    \hline
  \end{tabular}
  \caption{Essential set of hansl matrix functions}
  \label{tab:essential-matfuncs}
\end{table}

The following piece of code is meant to provide a concise example of
all the features mentioned above.

\begin{code}
# example: OLS using matrices

# fix the sample size
scalar T = 256

# construct vector of coefficients by direct imputation
matrix beta = {1.5, 2.5, -0.5} # note: row vector

# construct the matrix of independent variables
matrix Z = mnormal(T, cols(beta)) # built-in functions

# now construct the dependent variable: note the
# usage of the "dot" and transpose operators

matrix y = {1.2} .+ Z*beta' + mnormal(T, 1)

# now do estimation
matrix X = 1 ~ Z  # concatenation operator
matrix beta_hat1 = inv(X'X) * (X'y) # OLS by hand
matrix beta_hat2 = mols(y, X)       # via the built-in function
matrix beta_hat3 = X\y              # via matrix division

print beta_hat1 beta_hat2 beta_hat3
\end{code}

\section{Matrix pointers}
\label{sec:mat-pointers}

Hansl uses the ``by value'' convention for passing parameters to
functions. That is, when a variable is passed to a function as an
argument, what the function actually gets is a \emph{copy} of the
variable, which means that the value of the variable at the caller
level is not modified by anything that goes on inside the function.
But the use of pointers allows a function and its caller to cooperate
such that an outer variable can be modified by the function.

This mechanism is used by some built-in matrix functions to provide
more than one ``return'' value. The primary result is always provided
by the return value proper but certain auxiliary values may be
retrieved via ``pointerized'' arguments; this usage is flagged by
prepending the ampersand symbol, ``\texttt{\&}'', to the name of the
argument variable.

The \texttt{eigensym} function, which performs the eigen-analysis of
symmetric matrices, is a case in point. In the example below the first
argument $A$ represents the input data, that is, the matrix
whose analysis is required. This variable will not be modified in any
way by the function call. The primary result is the vector of
eigenvalues of $A$, which is here assigned to the variable
\texttt{ev}. The (optional) second argument, \texttt{\&V} (which may
be read as ``the address of \texttt{V}''), is used to retrieve the
right eigenvectors of $A$. A variable named in this way must be
already declared, but it need not be of the right dimensions to
receive the result; it will be resized as needed.
\begin{code}
matrix A = {1,2 ; 2,5}
matrix V
matrix ev = eigensym(A, &V)
print A ev V
\end{code}

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