202504022215 Matrices 矩阵

In Graphics, pervasively used to represent transformations.

• Translation, rotation, shear, scale

Matrices is array of numbers ($m\times n=m$ rows, n columns):

$$\left(\begin{array}{cc}1& 3\\ 5& 2\\ 0& 4\end{array}\right)$$

Matrix-Matrix Multiplication:

• # (number of) columns in A must = # rows in B $(M\times N)(N\times P)=(M\times P)$

$$\left(\begin{array}{cc}1& 3\\ 5& 2\\ 0& 4\end{array}\right)\left(\begin{array}{cccc}3& 6& 9& 4\\ 2& 7& 8& 3\end{array}\right)=\left(\begin{array}{cccc}9& 27& 33& 13\\ 19& 44& 61& 26\\ 8& 28& 32& 12\end{array}\right)$$

• Element (i, j) in the product is the dot product of row i from A and column from B 乘积矩阵的 (i,j) 元素为 A 矩阵的第 i 行向量和 B 矩阵的第 j 行向量点积的值

Matrix-Vecotr Multiplication:

• Treat vector as a column matrix $(m\times 1)$ 把向量视作 m 行 1 列的矩阵

$$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}-x\\ y\end{array}\right)$$

Transpose of a Matrix:

• Switch rows and columns $(ij\to ji)$

$${\left(\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right)}^T=\left(\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right)$$

• $(AB{)}^T=B^T{A}^T$

Identity Matrix and Inverses: 单位矩阵和矩阵的逆

$$I_{3\times 3}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

• $A{A}^{-1}=A^{-1}A=I$
• $(AB{)}^{-1}=B^{-1}{A}^{-1}$

Vector multiplication in Matrix form:

• Dot product
• Cross product