Matrices

In Graphics, pervasively used to represent transformations

• Translation, rotation, shear, scale

Matrices is array of numbers (m × n = m rows, n columns). Addition and multiplication by a scalar are trivial: element by element.

Matrix-Matrix Multiplication

• (number of) columns in A must = rows in B
  • (M x N) (N x P) = (M x P)
  • $\left(\begin{matrix}1 & 3\\5 & 2\\0 & 4\end{matrix}\right)\left(\begin{matrix}3 & 6 & 9 & 4\\2 & 7 & 8 & 3\end{matrix}\right) = \left(\begin{matrix}9 & 27 & 33 & 13\\19 & 44 & 61 & 26\\8 & 28 & 32 & 12\end{matrix}\right)$
• Element (i, j) in the product is the dot product of row i from A and column j from B
  • 比如上面矩阵相乘的结果，第二行第四列是 26，它等于矩阵 A 第二行 $\left(\begin{matrix}5 & 2\end{matrix}\right)$ 和矩阵 B 第四列 $\left(\begin{matrix}4\\3\end{matrix}\right)$ 的点积，即 26 = 5 × 4 + 2 × 3
• Properties
  • Non-Commutative (AB and BA are different in general)
  • Associative and distributive
    • (AB)C=A(BC)
    • A(B+C) = AB + AC
    • (A+B)C = AC + BC

Matrix-Vector Multiplication

• Treat vector as a column matrix (m×1)
• Key for transforming points (next lecture)

Transpose of a Matrix

• Switch rows and columns (ij -> ji)
  • $\left(\begin{matrix}1 & 2\\3 & 4\\5 & 6\end{matrix}\right)^T = \left(\begin{matrix}1 & 3 & 5\\2 & 4 & 6\end{matrix}\right)$
• Property
  • $(AB)^T = B^TA^T$

Identity Matrix and Inverses

• $I_{3 \times 3} = \left(\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right)$
• $AA^{-1} = A^{-1}A = I$
• $(AB)^{-1} = B^{-1}A^{-1}$

单位矩阵可以定义矩阵的逆。如果两个矩阵（不论顺序）相乘的结果是单位矩阵，那么这两个矩阵是互逆的。

Vector multiplication in Matrix form

• Dot product?
  • $\overrightarrow{a}\cdot\overrightarrow{b} = \overrightarrow{a}^T\overrightarrow{b} = \left(\begin{matrix}x_a & y_a & z_a\end{matrix}\right)\left(\begin{matrix}x_b\\y_b\\z_b\end{matrix}\right ) = (x_{a}x_{b} + y_{a}y_{b} + z_{a}z_{b})$
• Cross product?
  • $\overrightarrow{a}\times\overrightarrow{b} = A*\overrightarrow{b} = \left(\begin{matrix}0 & -z_a & y_a\\z_a & 0 & -x_a\\-y_a & x_a & 0\end{matrix}\right)\left(\begin{matrix}x_b\\y_b\\z_b\end{matrix}\right)$

矩阵形式的向量叉乘跟旋转的推导有关，后续章节再具体说。