LA@2@1@Linear equations and simple matrix equations have solution decision theorems

Matrix Equation Has Solution Judgment Theorem

Judgment of solutions of linear equations

• System of linear equations $A\mathbf{x}=The\; necessary\; and\; sufficient\; condition\; for\; b$ to have a solutionis its coefficient matrix A and augmented matrix$(A,b)$ have the same rank$R(A)=R(A,\mathbf{b})$,记$r=R\left(A\right)=R(A,\mathbf{b})$:

  • if $r=n$ has a system of equations with a unique solution
  • 若$r<n$ equations have multiple solutions

• For non-homogeneous linear equations, it is necessary to calculate $R(A),R(A,\mathbf{b})$

• For homogeneous linear equations only need to calculate $R(A)$

Specialization: Judgment of Solutions of Homogeneous Linear Equations

• This is a special case of a system of linear equations that has a solution, and the theorem can be further simplified

• A system of homogeneous linear equations $A\mathbf{x}=The\; case\; of\; 0$ homogeneous equations can be understood as$The\; elements\; in\; b$ are all 0

• It is easy to know that $Ax=0$ always$R\left(A\right)=R(\overline{A})=r$ , so the system of homogeneous linear equationsalways has a solution;

  • We only need to compute the coefficient matrix $A$ 's rank$R\left(A\right)$ to get$r$
  • if $r=n$ then the system of equations has a unique solution, and it isa zero solution
  • 若$r<A\; system\; of\; n$ equations has a non-zero solution

• Homogeneous linear equations have a solution determination theorem: homogeneous linear equations $A\mathbf{x}=$The necessary and sufficient condition for$0$$R(A)⩽n$;

  • The necessary and sufficient condition for zero solution (unique solution) is $R\left(A\right)=n$
  • The necessary and sufficient condition for non-zero solutions (multiple solutions) is $R\left(A\right)<n$;

Generalization: Matrix equation $AX=B$ has a solution

• BB here$B$ is a matrix of constant entries (no longer an augmented matrix of coefficient matrices)
• Theorem: Matrix equation $AX=$The necessary and sufficient condition for$B$$R(A)=R(A,B)$

  • Note here $X,B$ is not necessarily a vector, it may be a matrix with multiple rows and columns

  • Refer to Tongji Line Code v6@p76@Theorem 6

prove

• A $A,X,B$ are m × nm\times{n}respectively$m\times n$,$n\times l$,$m\times$matrix of$l$

• Block X and B by column:

  • $X$=$({\mathbf{x}}_1,{\mathbf{x}}_2,\cdots x_l)$,
  • $B$=$({\mathbf{b}}_1,{\mathbf{b}}_2,\cdots b_l)$

• Matrix equation $AX=B$ is equivalentto$l$ vectorequations(system of linear equations)

• $AX=A({\mathbf{x}}_1,{\mathbf{x}}_2,\cdots x_l)$=$(A{\mathbf{x}}_1,A{\mathbf{x}}_2,\cdots A{x}_l)$

• All $AX=B$等价于$(A{x}_1,A{x}_2,\cdots A{x}_l)$=$({\mathbf{b}}_1,{\mathbf{b}}_2,\cdots b_l)$

  • 又等价于$A{\mathbf{x}}_i=b_i(i=1,2,\cdots ,l)$ a total of$l$ linear equations
  • What these linear equations have in common is the same coefficient matrix $A$ , which means thellThe ranks of the coefficient$matrices\; of\; the\; l$ linear equations and the original matrix equationsare equal, this conclusion is very important
  • The position number matrix and the constant term matrix are relatively independent

• 设$R(A)=r$ , andAAThe row echelonof$A$ is$\tilde{A}$,则$\tilde{A}$there is $r$ non-zero lines, and$\tilde{A}$After $m-r$ line with all zeros

• $(A,B)$=$(A,{\mathbf{b}}_1,b_2,\cdots b_l)$ $\stackrel{r}{\sim }$ $(\tilde{A},\widetilde{b_1},\cdots ,\widetilde{b_l})$

  • where $\tilde{A}$Yes AAThe row echelon formmatrixof$A$
  • And the vector $\widetilde{{\mathbf{b}}_1},\cdots ,\widetilde{b_l}$是$b_1,b_2,\cdots b_l$与$A\stackrel{r}{\sim }\tilde{A}$The result after performing the same row transformation, namely $\widetilde{b_i}$does not represent a row echelon matrix

• will be equivalent to the $The\; elementary\; row\; transformation\; of\; the\; augmented\; matrix\; of\; i$ linear equations is a row echelon matrix:$(A,b_i)$ $\stackrel{r}{\sim }$ $(\tilde{A},\widetilde{b_i})$,$(i=1,2,\cdots ,l)$

• $AX=B$有解$\iff$ $A{\mathbf{x}}_i={\mathbf{b}}_i$ $(i=1,2,\cdots ,l)$ have a solution

  • $\iff$ $R(A,b_i)$=$R(A)=r$,$(i=1,2,\cdots ,l)$
  • $\iff$ $\widetilde{b_i}$After $m-r$ components (units) are all 0$(i=1,2,\cdots ,l)$
    • Because, if after $m-There\; are\; non-zero\; elements\; in\; r$ elements, which will cause$R(A,{\mathbf{b}}_i)>R(A)$,导致$A{\mathbf{x}}_i=b_i$No solution
    • And its former $The\; value\; of\; r$ elements will not affect$R(A,b_i)$= R ( A ) R(A) We don't care about the establishment of$R$$($$A$$)$
  • $\iff$ 矩阵$(\widetilde{{\mathbf{b}}_1},\cdots ,\widetilde{b_l})$ after$m-R$ line is all 0;
  • $\iff$ row echelon matrix$\tilde{D}$=$(\tilde{A},\widetilde{b_1},\cdots ,\widetilde{b_l})$ after$m-R$ line is all 0
  • $\iff$ $R(\tilde{D})⩽m-(m-r)=r$ , and because$\tilde{D}$contains $\tilde{A}$, so $R\left(\tilde{A}\right)=r⩽R\left(\tilde{D}\right)$
  • $\iff$ $R(\tilde{D})=r$
  • $\iff R(A,B)=R(A)$

• Therefore, if $AX=B$有解,则$R(A,B)=R\left(A\right)$

inference

• Young $AX=B$ has a solution, then$R(B)⩽R(A,B)=R\left(A\right)$ , so$R\left(B\right)⩽R\left(A\right)$ , that is,the rank of the constant term matrix is ​​less than the rank of the coefficient matrix
• against $AX=Both\; sides\; of\; B$ take the transpose operation at the same time, there is$X^T{A}^T=B^T$ , in the same way$R(B^T)⩽R(X^T)$,即$R(B)⩽R(X)$
• Finally, $R\left(B\right)⩽\min (R(A),R(X))$