-矩陣
任給數(shù)域P上n維空間V上線性變換,已定義過P上的多項式,即![\forall f(\lambda)\in P[\lambda]](https://math.jianshu.com/math?formula=%5Cforall%20f(%5Clambda)%5Cin%20P%5B%5Clambda%5D)
,%3Da_k%5Clambda%5Ek%2Ba_%7Bk-1%7D%5Clambda%5E%7Bk-1%7D%2B%5Ccdots%2Ba_1%5Clambda%2Ba_0)
,稱%3Da_k%5Cmathscr%7BA%7D%5Ek%2Ba_%7Bk-1%7D%5Cmathscr%7BA%7D%5E%7Bk-1%7D%2B%5Ccdots%2Ba_1%5Cmathscr%7BA%7D%2Ba_0%5Cmathscr%7BE%7D)
為P上的多項式,其中
為V上恒等變換,)
仍為V上線性變換
定義：對P上任意
-矩陣![K(\lambda)=(a_{ij}(\lambda))_{m\times l},a_{ij}(\lambda)\in P[\lambda]](https://math.jianshu.com/math?formula=K(%5Clambda)%3D(a_%7Bij%7D(%5Clambda))_%7Bm%5Ctimes%20l%7D%2Ca_%7Bij%7D(%5Clambda)%5Cin%20P%5B%5Clambda%5D)
,令%3D(a_%7Bij%7D(%5Cmathscr%7BA%7D))_%7Bm%5Ctimes%20l%7D)
,稱為P上的一個-矩陣
例：
%3D%5Cbegin%7Bpmatrix%7D%5Clambda%5E2%2B1%26%5Clambda-1%5C%5C3%26%5Clambda%5E3%2B%5Clambda%5Cend%7Bpmatrix%7D)
,%3D%5Cbegin%7Bpmatrix%7D%5Cmathscr%7BA%7D%5E2%2B%5Cmathscr%7BE%7D%26%5Cmathscr%7BA%7D-%5Cmathscr%7BE%7D%5C%5C3%5Cmathscr%7BE%7D%26%5Cmathscr%7BA%7D%5E3%2B%5Cmathscr%7BA%7D%5Cend%7Bpmatrix%7D)
若設%3D%5Clambda%20E-B)
,稱為B的特征矩陣,設_%7Bn%5Ctimes%20n%7D)
,則
%3D%5Clambda%20E-B%3D%5Cbegin%7Bpmatrix%7D%5Clambda-b_%7B11%7D%26-b_%7B12%7D%26%5Ccdots%26-b_%7B1n%7D%5C%5C%20-b_%7B21%7D%26%5Clambda-b_%7B22%7D%26%5Ccdots%26-b_%7B2n%7D%5C%5C%20%5Cvdots%26%5Cvdots%26%20%26%5Cvdots%5C%5C%20-b_%7Bn1%7D%26-b_%7Bn2%7D%26%5Ccdots%26%5Clambda-b_%7Bnn%7D%5Cend%7Bpmatrix%7D)
%3D%5Cbegin%7Bpmatrix%7D%5Cmathscr%7BA%7D-b_%7B11%7D%5Cmathscr%7BE%7D%26-b_%7B12%7D%5Cmathscr%7BE%7D%26%5Ccdots%26-b_%7B1n%7D%5Cmathscr%7BE%7D%5C%5C%20-b_%7B21%7D%5Cmathscr%7BE%7D%26%5Cmathscr%7BA%7D-b_%7B22%7D%5Cmathscr%7BE%7D%26%5Ccdots%26-b_%7B2n%7D%5Cmathscr%7BE%7D%5C%5C%20%5Cvdots%26%5Cvdots%26%20%26%5Cvdots%5C%5C%20-b_%7Bn1%7D%5Cmathscr%7BE%7D%26-b_%7Bn2%7D%5Cmathscr%7BA%7D%26%5Ccdots%26%5Cmathscr%7BA%7D-b_%7B1n%7D%5Cmathscr%7BE%7D%5Cend%7Bpmatrix%7D%E2%80%8B)
矩陣運算及行列式定義中只用到元素的加法和乘法,而
的多項式有加法和乘法
與-矩陣一樣,-矩陣也有加法、乘法及"數(shù)量"乘法(用的多項式作為元素與-矩陣,或與數(shù)字矩陣作"數(shù)量"乘法),也有與數(shù)字矩陣驶悟、-矩陣類似的運算性質,也可定義-矩陣的行列式,也有與數(shù)字行列式相同的性質,如行列式乘法定理胡野、伴隨矩陣的存在性等
用-矩陣運算可將)
表為
%3D%5Cbegin%7Bpmatrix%7D%5Cmathscr%7BA%7D-b_%7B11%7D%5Cmathscr%7BE%7D%26-b_%7B12%7D%5Cmathscr%7BE%7D%26%5Ccdots%26-b_%7B1n%7D%5Cmathscr%7BE%7D%5C%5C%20-b_%7B21%7D%5Cmathscr%7BE%7D%26%5Cmathscr%7BA%7D-b_%7B22%7D%5Cmathscr%7BE%7D%26%5Ccdots%26-b_%7B2n%7D%5Cmathscr%7BE%7D%5C%5C%20%5Cvdots%26%5Cvdots%26%20%26%5Cvdots%5C%5C%20-b_%7Bn1%7D%5Cmathscr%7BE%7D%26-b_%7Bn2%7D%5Cmathscr%7BA%7D%26%5Ccdots%26%5Cmathscr%7BA%7D-b_%7B1n%7D%5Cmathscr%7BE%7D%5Cend%7Bpmatrix%7D%3D%5Cmathscr%7BA%7DE%3D%5Cmathscr%7BE%7DB)
對![f(\lambda),g(\lambda)\in P[x]](https://math.jianshu.com/math?formula=f(%5Clambda)%2Cg(%5Clambda)%5Cin%20P%5Bx%5D)
,令%3Df(%5Clambda)%2Bg(%5Clambda)%2Cp(%5Clambda)%3Df(%5Clambda)g(%5Clambda))
則%3Df(%5Cmathscr%7BA%7D)%2Bg(%5Cmathscr%7BA%7D)%2Cp(%5Cmathscr%7BA%7D)%3Df(%5Cmathscr%7BA%7D)g(%5Cmathscr%7BA%7D))
同時,對-矩陣,_%7Bm%5Ctimes%20k%7D%2CL(%5Clambda)_%7Bm%5Ctimes%20k%7D)
,若%3DK(%5Clambda)%2BL(%5Clambda))
則%3DK(%5Cmathscr%7BA%7D)%2BL(%5Cmathscr%7BA%7D))
對_%7Bm%5Ctimes%20s%7D%2CL(%5Clambda)_%7Bs%5Ctimes%20k%7D)
,若%3DK(%5Clambda)L(%5Clambda))
則%3DK(%5Cmathscr%7BA%7D)L(%5Cmathscr%7BA%7D))
對V上的一個線性變換,設它在基
下的矩陣為
%3D(%5Cvarepsilon_1%2C%5Cvarepsilon_2%2C%5Ccdots%2C%5Cvarepsilon_n)A')
設)
則
即
注：可用-矩陣的形式表達式寫出
矩陣A稱為在基
下的左矩陣
在同一基下的矩陣A'和左矩陣A互為轉置關系,有相同特征多項式
在不同基下的矩陣是相似的,它們對應的轉置也相似,即在不同基下的左矩陣也相似,故在任何基下的矩陣及左矩陣的特征多項式都相等,都是的特征多項式
或%3D%5Cbegin%7Bpmatrix%7D%5Cvarepsilon_1%5C%5C%5Cvarepsilon_2%5C%5C%5Cvdots%5C%5C%5Cvarepsilon_n%5Cend%7Bpmatrix%7D%3DO)
注：左端為形式寫法,是-矩陣與向量元素的矩陣的"乘法",不是-矩陣的乘法
-矩陣%3D(k_%7Bij%7D(%5Cmathscr%7BA%7D))_%7Bm%5Ctimes%20n%7D)
及向量矩陣
,
稱%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7Dk_%7B11%7D%5Cxi_1%2Bk_%7B12%7D%5Cxi_2%2B%5Ccdots%2Bk_%7B1n%7D%5Cxi_n%5C%5C%20k_%7B12%7D%5Cxi_1%2Bk_%7B22%7D%5Cxi_2%2B%5Ccdots%2Bk_%7B2n%7D%5Cxi_n%5C%5C%20%5Cvdots%5C%5C%20k_%7Bn1%7D%5Cxi_1%2Bk_%7Bn2%7D%5Cxi_2%2B%5Ccdots%2Bk_%7Bnn%7D%5Cxi_n%5Cend%7Bpmatrix%7D%E2%80%8B)
為-矩陣的形式寫法,與數(shù)字矩陣的形式寫法一樣,有性質：
1._%7Bm%5Ctimes%20s%7DL(%5Cmathscr%7BA%7D)_%7Bs%5Ctimes%20k%7D)%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_k%5Cend%7Bpmatrix%7D%3DK(%5Cmathscr%7BA%7D)%5Cbegin%7Bbmatrix%7DL(%5Cmathscr%7BA%7D)%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%5Cend%7Bbmatrix%7D)
2.![[K(\mathscr{A})_{m\times k}+L(\mathscr{A})_{m\times k}]\begin{pmatrix}\xi_1\\\xi_2\\\vdots\\\xi_n\end{pmatrix}=K(\mathscr{A})\begin{pmatrix}\xi_1\\\xi_2\\\vdots\\\xi_n\end{pmatrix}+L(\mathscr{A})\begin{pmatrix}\xi_1\\\xi_2\\\vdots\\\xi_n\end{pmatrix}](https://math.jianshu.com/math?formula=%5BK(%5Cmathscr%7BA%7D)_%7Bm%5Ctimes%20k%7D%2BL(%5Cmathscr%7BA%7D)_%7Bm%5Ctimes%20k%7D%5D%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%3DK(%5Cmathscr%7BA%7D)%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%2BL(%5Cmathscr%7BA%7D)%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D)
3.設是一個-多項式,則
![[f(\mathscr{A})K(\mathscr{A})]\begin{pmatrix}\xi_1\\\xi_2\\\vdots\\\xi_n\end{pmatrix}=K(\mathscr{A})\begin{bmatrix}K(\mathscr{A})\begin{pmatrix}\xi_1\\\xi_2\\\vdots\\\xi_n\end{pmatrix}\end{bmatrix}](https://math.jianshu.com/math?formula=%5Bf(%5Cmathscr%7BA%7D)K(%5Cmathscr%7BA%7D)%5D%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%3DK(%5Cmathscr%7BA%7D)%5Cbegin%7Bbmatrix%7DK(%5Cmathscr%7BA%7D)%5Cbegin%7Bpmatrix%7D%5Cxi_1%5C%5C%5Cxi_2%5C%5C%5Cvdots%5C%5C%5Cxi_n%5Cend%7Bpmatrix%7D%5Cend%7Bbmatrix%7D)
注：-矩陣的形式寫法與數(shù)字矩陣的形式有不同性質
例：對可逆的數(shù)字矩陣
及一組基,
作
則
仍是一組基
對可逆的-矩陣_%7Bn%5Ctimes%20n%7D)
即存在-矩陣_%7Bn%5Ctimes%20n%7D)
使K(%5Cmathscr%7BA%7D)%3DK(%5Cmathscr%7BA%7D)L(%5Cmathscr%7BA%7D)%3DE)
對基作形式寫法
%5Cbegin%7Bpmatrix%7D%5Cvarepsilon_1%5C%5C%5Cvarepsilon_2%5C%5C%5Cvdots%5C%5C%5Cvarepsilon_n%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%5Ceta_1%5C%5C%5Ceta_2%5C%5C%5Cvdots%5C%5C%5Ceta_n%5Cend%7Bpmatrix%7D%E2%80%8B)
可能
不是基,可能有
例：設
%3D%5Cbegin%7Bpmatrix%7D1%26-%5Clambda%2B3%26-5%5C%5C0%261%26-1%5C%5C0%260%261%5Cend%7Bpmatrix%7D)
%7C%3D1)
,故)
是可逆-矩陣
設L(%5Clambda)%3DL(%5Clambda)Q(%5Clambda)%3DE)
則L(%5Cmathscr%7BA%7D)%3DL(%5Cmathscr%7BA%7D)Q(%5Cmathscr%7BA%7D)%3D%5Cmathscr%7BE%7D)
即)
是可逆矩陣
%5Cbegin%7Bpmatrix%7D%5Cvarepsilon_1%5C%5C%5Cvarepsilon_2%5C%5C%5Cvarepsilon_3%5Cend%7Bpmatrix%7D)
%5Cvarepsilon_2-5%5Cvarepsilon_3%5C%5C%5Cvarepsilon_2-%5Cvarepsilon_3%5C%5C%5Cvarepsilon_3%5Cend%7Bpmatrix%7D)
%2B3%5Cvarepsilon_2)-5%5Cvarepsilon_3%5C%5C%5Cvarepsilon_2-%5Cvarepsilon_3%5C%5C%5Cvarepsilon_3%5Cend%7Bpmatrix%7D)
即
,
不能為基
