Multi-line Equations and Matrices

Chapter 4 covered the building blocks. This lesson combines them into the multi-line equations and matrices that research papers and textbooks demand.

Multi-line Equations

Most real derivations span multiple lines. The amsmath package provides several environments for this.

The align Environment

The workhorse of multi-line math. Lines are aligned at the & character and separated by \\:

\begin{align}
  (x+1)^3 &= (x+1)(x+1)^2 \\
           &= (x+1)(x^2+2x+1) \\
           &= x^3+3x^2+3x+1
\end{align}

Rendered result:

$$ \begin{aligned} (x+1)^3 &= (x+1)(x+1)^2 \\ &= (x+1)(x^2+2x+1) \\ &= x^3+3x^2+3x+1 \end{aligned} $$

Each line gets its own equation number. To suppress numbering on specific lines, add \nonumber before \\. To suppress all numbering, use align*.

The gather Environment

When lines don’t share an alignment point, use gather. Each line is centered independently:

\begin{gather}
  x^2 + y^2 = r^2 \\
  e^{i\theta} = \cos\theta + i\sin\theta \\
  \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}
\end{gather}

Rendered result:

$$ x^2 + y^2 = r^2 $$ $$ e^{i\theta} = \cos\theta + i\sin\theta $$ $$ \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} $$

The multline Environment

For a single equation that’s too long for one line. The first line is left-aligned, the last is right-aligned, and middle lines are centered:

\begin{multline}
  p(x) = x^8 + 3x^7 - 2x^6 + 5x^5 \\
    - 7x^4 + 4x^3 + 11x^2 \\
    - 9x + 13
\end{multline}

Rendered result:

$$ p(x) = x^8 + 3x^7 – 2x^6 + 5x^5 – 7x^4 + 4x^3 + 11x^2 – 9x + 13 $$

Choosing the Right Environment

• Environment: equation | When to Use: Single equation, numbered
• Environment: align | When to Use: Multi-line with alignment (derivations, systems)
• Environment: gather | When to Use: Multi-line without alignment (unrelated equations)
• Environment: multline | When to Use: One long equation broken across lines
• Environment: split | When to Use: Multi-line inside a single equation (one number)
• Environment: aligned | When to Use: Aligned block inside another math environment

  The split Environment

When you want aligned lines but only one equation number, wrap split inside equation:

\begin{equation}
\begin{split}
  \nabla \times \vec{E}
    &= -\frac{\partial \vec{B}}{\partial t} \\
  \nabla \times \vec{B}
    &= \mu_0 \vec{J}
    + \mu_0 \varepsilon_0
    \frac{\partial \vec{E}}{\partial t}
\end{split}
\end{equation}

Rendered result:

$$ \begin{aligned} \nabla \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{B} &= \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \end{aligned} $$

The entire block gets a single equation number on the right, unlike align where each line is numbered separately.

Matrices

The amsmath package provides matrix environments that differ only in their delimiters.

Matrix Types

% No delimiters
$\begin{matrix}
  a & b \\ c & d
\end{matrix}$

% Parentheses
$\begin{pmatrix}
  a & b \\ c & d
\end{pmatrix}$

% Brackets
$\begin{bmatrix}
  a & b \\ c & d
\end{bmatrix}$

% Braces
$\begin{Bmatrix}
  a & b \\ c & d
\end{Bmatrix}$

% Vertical bars (determinant)
$\begin{vmatrix}
  a & b \\ c & d
\end{vmatrix}$

% Double bars (norm)
$\begin{Vmatrix}
  a & b \\ c & d
\end{Vmatrix}$

Rendered results:

No delimiters:

$$ \begin{matrix} a & b \\ c & d \end{matrix} $$

Parentheses:

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$

Brackets:

$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

Braces:

$$ \begin{Bmatrix} a & b \\ c & d \end{Bmatrix} $$

Vertical bars (determinant):

$$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} $$

Double bars (norm):

$$ \begin{Vmatrix} a & b \\ c & d \end{Vmatrix} $$

Large Matrices with Dots

For matrices where you need to show a general pattern, use dot commands:

<!--M14-->

Rendered result:

$$ A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$

The three dot commands are:

• \cdots for horizontal dots (centered)
• \vdots for vertical dots
• \ddots for diagonal dots

Augmented Matrices

For augmented matrices (common in linear algebra), use the array environment with a vertical bar in the column specification:

<!--M15-->

Rendered result:

$$ \left[\begin{array}{ccc|c} 1 & 0 & 2 & 3 \\ 0 & 1 & -1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

The {ccc|c} column specification means three centered columns, a vertical line, then one more centered column. The \left[ and \right] provide the outer brackets that scale automatically.