[advanced mathematics] elementary transformation of matrix and determinant

Elementary transformation of matrix and determinant

In Linear Algebra , Elementary transformation is the most important operation , However, the elementary transformation of matrix and determinant are often confused , The purpose of this paper is to clarify these concepts ： matrix 、 determinant 、 Elementary transformation 、 Elementary matrix 、 Elementary transformation of matrix 、 Elementary transformation of determinant .

One 、 Matrices and determinants

• The matrix is a Number table , Usually enclosed in brackets ：

$${\mathbf{A}}_{3\times 4}=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 2& 0& 2\\ 0& 0& 3& 3\end{array}\right]$$

There's a 3 That's ok 4 Columns of the matrix .

• Determinant is a Count , Through to Matrix The number obtained by operation ：

$$det\mathbf{A}=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right|=2$$

The number of rows and columns of a matrix may not be equal , The number of rows and columns of a row column formula must be equal .

Two 、 Elementary transformation and elementary matrix

The so-called elementary transformation is the three most basic transformations of matrix , Elementary matrix is the matrix representation corresponding to these three transformations .

Elementary transformation is divided into row elementary transformation and column elementary transformation . The difference between the two lies in the matrix being calculated $\mathbf{A}$ Left or right .

Three elementary transformations , And the corresponding elementary matrix ：

1. In exchange for ： take i Row sum j Line exchange , The corresponding elementary matrix is the unit matrix i Row sum j Line exchange
   $${\mathbf{E}}_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$$

2. multiplier ： take i Multiply each number of rows by c,
   $${\mathbf{E}}_i(c)=\left[\begin{array}{ccc}1& 0& 0\\ 0& c& 0\\ 0& 0& 1\end{array}\right]$$

3. Multiply and add ： take i Multiply each number of rows by c Add to j That's ok ,
   $${\mathbf{E}}_{ij}(c)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& c& 1\end{array}\right]$$

Now let's demonstrate ：
$${\mathbf{E}}_{ij}\mathbf{A}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\times \left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 2& 0& 2\\ 0& 0& 3& 3\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& 3& 3\\ 0& 2& 0& 2\end{array}\right]$$
If the elementary matrix is placed in A To the right of is the column transformation .

3、 ... and 、 Elementary transformation of matrix and determinant

1. The elementary transformation of matrix is to keep Equivalence relation

2. The elementary transformation of determinant is to keep Equivalence relations

The elementary transformation method of matrix has been made clear above .

Elementary transformation rules of determinant ：

1. In exchange for ：$det{\mathbf{E}}_{ij}\mathbf{A}=-det\mathbf{A}$
2. multiplier ：$det{\mathbf{E}}_i(c)\mathbf{A}=cdet\mathbf{A}$
3. Multiply and add ：$det{\mathbf{E}}_{ij}(c)\mathbf{A}=det\mathbf{A}$

Four 、 Number multiplication and multiple multiplication

• For matrices ：

Number multiplication ：$k\mathbf{A}$ , the $\mathbf{A}$ Each element in is multiplied by k

multiplier ：${\mathbf{E}}_i(c)\mathbf{A}$, the $\mathbf{A}$ Of the i Multiply each element of the row by c

• For determinants ：

Number multiplication ：$det(k{A}_{n\times n})=k^{n}det(A)$

multiplier ：$det({\mathbf{E}}_i(c)A_{n\times n})=kdet(A)$