matrix.tex

% matrices and vectors for linera algebra
% not intended to be compiled alone


% for vectors in 3D

% addition of vectors
\vec{u} & = (u_x, u_y, u_z) \\
\vec{v} & = (v_x, v_y, v_z) \\
\vec{u} + \vec{v} & = (u_x + v_x, u_y + v_y, u_z + v_z)

% subtraction of vectors
\vec{u} & = (u_x, u_y, u_z) \\
\vec{v} & = (v_x, v_y, v_z) \\
\vec{u} - \vec{v} & = (u_x - v_x, u_y - v_y, u_z - v_z)

% vector dot product
\vec{u} & = (u_x, u_y, u_z) \\
\vec{v} & = (v_x, v_y, v_z) \\
\vec{u} \cdot \vec{v} & = (u_x * v_x) + (u_y * v_y) + (u_z * v_z)

% vector magnitude
\vec{u} & = (u_x, u_y, u_z) \\
| \vec{u} | & = \sqrt{(u_x)^2 + (u_y)^2 + (u_z)^2}

% vector cross product
\vec{u} & = (u_x, u_y, u_z) \\
\vec{v} & = (v_x, v_y, v_z) \\
\vec{u} \times \vec{v} & = ((u_y * v_z) - (u_z * v_y), (u_z * v_x) - (u_x * v_z), (u_x * v_y) - (u_y * v_x))

% several sample matrices
\begin{align*}
  \left[\begin{array}{ccc}
    -1 & 0 & 0 \\
    0 & -1 & 0 \\
    0 & 0 & -1
  \end{array}\right]
\end{align*}

\begin{align*}
  \left[\begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & -1
  \end{array}\right]
\end{align*}

\begin{align*}
  \left[\begin{array}{ccc}
    -1 & 0 & 0 \\
    0 & -1 & 0 \\
    0 & 0 & 1
  \end{array}\right]
\end{align*}

% matrix multiplication for 2x2 matrices
\begin{align*}
  \left[\begin{array}{rr}
    a & b \\
    c & d
  \end{array} \right]
  \left[\begin{array}{rr}
    e & f \\
    g & h
  \end{array} \right]
  & = 
  \left[\begin{array}{rr}
    a*e + b*g & a*f + b*h \\
    c*e + d*g & c*f + d*h
  \end{array} \right]
\end{align*}