[TA frost wolf \u may- hundred talents plan] 1.2.2 matrix calculation

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1.2.2 Matrix computing

1.2.2.1 matrix

Definition ： One m x n The matrix of is a matrix of m That's ok n A rectangular array of column elements .
In graphics, we usually operate on the square matrix , Two dimensional second order , Three dimensional third order , Three dimensional band translation is fourth order .
There are other examples ：

• Unit matrix
• Zero matrix Zero matrices of different sizes are different , Identity matrix is the same .

1.2.2.2 The addition and subtraction of matrices

Premise ： Homomorphic matrices
The addition and subtraction of matrix is the addition of corresponding positions .

1.2.2.3 The multiplication of a matrix

Multiply each position by a scalar

1.2.2.4 Multiplication of matrices

Need to meet a certain degree of legitimacy
m x n The matrix of can only be combined with n x p matrix multiplication
The matrix size after multiplication must be m x p
Geometric meaning ：

• matrix x A matrix can be thought of as a scaling overlay rotation , Or a form of transformation
• matrix x vector Finally, we get a new vector or vertex coordinate

Compound transformation in matrix ：

There can be two calculation orders ：

Satisfy the operation law ：

Commonly used transformation matrix ：

Coordinate transformation in three-dimensional space ：

Unity in , Rotation in three-dimensional space follows z、x、y In the order of

1.2.2.5 The transpose of the matrix

The transpose of a matrix has the following properties ：

unity When matrix operation is performed on each point of the model in , The operation on the fourth column cannot achieve the purpose of Translation , However, the corresponding effect can be achieved by operating the fourth line at the corresponding position , The reason is Unity Matrix stored procedure , The whole is transposed , So in the subsequent code process , It is suggested to transpose the translation matrix and then perform additional operations .

1.2.2.6 Inverse matrix

Calculation of inverse matrix ：

Properties of inverse matrix ：

summary

Various geometric meanings in matrix calculation ：

1. Addition and subtraction ： The change of unit vector
2. Number multiplication ： Scaling of unit vectors
3. Matrix multiplication ： The zoom 、 rotate 、 Displacement （ High dimension required ）、 Oblique cutting 、 Mirror symmetry , The tensile , Various shaft rotation, etc
4. Transposition ： The positions of rows and columns are interchangeable
5. The matrix of the inverse ： Changes to a matrix , It can be restored by multiplying the inverse matrix

The matrix formula ：

1. Addition and subtraction ： The shape of the matrix is exactly the same , Add the corresponding positions
2. Number multiplication ： Multiply each position of the matrix by the scalar
3. Matrix multiplication ： From right to left . When two matrices are multiplied , The rows of the matrix on the left remain unchanged , Traverse the columns of the matrix on the right in turn and multiply , Fill in the same rows of the matrix on the left in order , Until the matrix traversal on the right is completed , Then jump to the next row of the matrix on the left , Repeat the above operation , Until the last row of the matrix on the left . therefore , The shape of the two matrices must satisfy that the left side is ：m x n, On the right is ：n x p, That is, the number of columns of the left matrix is equal to the number of rows of the right matrix .
4. The matrix of the inverse ： The inverse of a matrix is multiplied by the inverse of a matrix to form an identity matrix