LaTeX – Multiline equations, systems and matrices ? Machine Intelligence ?

Series on Blogging with LaTeX

This is the 3rd post in the series. Previous ones:

• Basics and overview
• Use of mathematical symbols in formulas and equations

Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup.
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Multiline Equations

You can present equations with several lines, using the array statement. Inside its declaration you must :

• Define the number of columns
• Define column alignment
• Define column indentation
• Indicate column separator with & symbol &

Example: {lcr} means: 3 columns with indentations respectively left, center and right

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array}{rcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array}{rcl} f: R^3 & \to & R \\ (x,y,z) & \to & x + y + z \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array} {lcl} f(x) & = & (a+b)^2 \\ & = & a^2+2ab+b^2 \end{array}

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Case definitions

Used when a definition have two or more cases. Use the case statement. Notice that the spaces after the instances of if were included inside the mbox declarations.

f(n) = \begin{cases} n/2, & \mbox{if } n\mbox{ is even} \\ 3n+1, & \mbox{if } n\mbox{ is odd} \end{cases}

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Simultaneous Equations

Here we have a very simple application of the case statement.

\begin{cases} 3x + 5y + z \\ 7x – 2y + 4z \\ -6x + 3y + 2z \end{cases}

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Matrices

Matrices can be assembled by using the array statement, like in this example:

\left| \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array} \right|

Matrix frames are provided by \left and \right. If you suppress these statements, it will be displayed like:

\begin{array}{cc} A & B \\ C & D \end{array}

However, there is another statement, the matrix declaration, slightly easier to use:

\begin{matrix} x & y \\ z & v \end{matrix}

The frames of the matrix can be displayed in several forms, by just changing the matrix declaration to vmatrix, Vmatrix, bmatrix, Bmatrix or pmatrix, as shown ahead:

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

\begin{bmatrix} x & y \\ z & v \end{bmatrix}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

As a final example, let us construct more complicated matrices and matricial expressions:

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

Now, we will show a first matricial expression:

\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] =\begin{bmatrix} A & B \\ C & D \end{bmatrix} \times \left[\begin{array}{c} y_1 \\ y_2 \end{array} \right]

and, here is another example, now using two kinds of vectors:

\begin{bmatrix} xz & xw \\ yz & yw \end{bmatrix} = \left[\begin{array}{c} x \\ y \end{array} \right] \times \left[\begin{array}{cc} z & w \end{array} \right]