matrix
A matrix is a set of numbers with several rows and columns. For example, the following is a matrix with three rows and two columns.
( 1 3 5 2 0 4 ) \begin{pmatrix} 1 & 3 \\ 5 & 2 \\ 0 &4 \end{pmatrix} ⎝⎛150324⎠⎞
Matrix operations
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Matrix multiplication
The requirement for matrix multiplication is that the number of columns of the first matrix must be equal to the number of rows of the second matrix, m × n m\times n m×n 的矩阵用 n × p n\times p n×Multiply the matrices of p to finally get m × p m\times p m×The matrix of p.
The result of the new matrix, such as the value of the second row and third column of the new matrix, is equal to the second row of the first matrix multiplied by the third column of the second matrix, and then added Get the sum.
( 1 3 5 2 0 4 ) ( 3 6 9 4 2 7 8 3 ) = ( 9 27 33 13 19 44 61 26 8 28 32 12 ) \begin{pmatrix} 1 & 3 \\ 5 & 2 \\ 0 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 & 4 \\ 2 & 7 & 8 & 3 \end{pmatrix} = \begin{pmatrix} 9 & 27 & 33 & 13 \\ 19 & 44 & 61 & 26 \\ 8 & 28 & 32 & 12 \end{pmatrix} ⎝⎛150324⎠⎞(32679843)=⎝⎛9198274428336132132612⎠⎞
Multiply a matrix and a vector, and the vector can be regarded as a m × 1 m\times 1 m×1's size. -
transpose of matrix
The transpose of the matrix is to swap the rows and columns, such as:
( 1 2 3 4 5 6 ) T = ( 1 3 5 2 4 6 ) \begin{pmatrix } 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix} ^ T = \begin{pmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{pmatrix } ⎝⎛135246⎠⎞T=(123456)
Properties of matrix transpose:
The transpose of matrix multiplication is equivalent to the inverse multiplication of the two transposed matrices:
( A B ) T = B T A T (AB)^T=B^TA^T (AB)T=BTAT -
Identity matrix
The identity matrix also has many dimensions, but multiplying the identity matrix with other matrices will not perform any operation and is still equal to the original matrix.
I 3 × 3 = ( 1 0 0 0 1 0 0 0 1 ) I_{3\times3}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} I3×3=⎝⎛100010001⎠⎞
But the identity matrix can lead to the properties of the inverse matrix:
If a matrix multiplied by another matrix equals the identity matrix, then we say that the two matrices are inverse matrices of each other.
A A − 1 = A − 1 A = I AA^{-1}=A^{-1}A=I AA−1=A−1A=I
Inverse square property:
( A B ) − 1 = B − 1 A − 1 (AB)^{-1}= B^{-1}A^{-1} (AB)−1=B−1A−1 -
Vector multiplication is expressed in matrix form
- 点乘
a ⃗ ⋅ b ⃗ = a ⃗ T b ⃗ = ( x a y a z a ) ( x b y b z b ) = ( x a x b + y a y b + z a z b ) \vec{a} \cdot \vec{b} = \vec{a}^T \vec{b} = \begin{pmatrix} x_a & y_a & z_a \end{pmatrix} \begin{pmatrix} x_b \\ y_b \\ z_b \end{pmatrix}=(x_ax_b+y_ay_b+z_az_b) a⋅b=aTb=(xaandaWitha)⎝⎛xbandbWithb⎠⎞=(xaxb+andaandb+WithaWithb) - 叉乘
a ⃗ × b ⃗ = A ⃗ ∗ b ⃗ = ( 0 − z a y a z a 0 − x a − y a x a 0 ) ( x b y b z b ) \vec{a} \times \vec{b} = \vec{A}^* \vec{b} =\begin{pmatrix} 0 & -z_a & y_a \\ z_a & 0 & -x_a \\ -y_a & x_a & 0 \end{pmatrix} \begin{pmatrix} x_b \\ y_b \\ z_b \end{pmatrix} a×b=A∗b=⎝⎛0Witha−ya−za0xaanda−xa0⎠⎞⎝⎛xbandbWithb⎠⎞
这り的 A ⃗ ∗ \vec{A}^* A∗ is a matrix, not A ⃗ ∗ b ⃗ \vec{A}*\vec{b} A∗b。