线性代数-行列式性质

2019-11-05  本文已影响0人  d0dcdf2fcfdc


D =\left|\begin{array}{ccc}a_{ij} \end{array}\right|=\left|\begin{array}{ccc}a_{11} & a_{12} & ··· & a_{1n} \\a_{21} & a_{22} & ··· & a_{2n} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & x_{nn} \end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n}) + \tau(P_{1}P_{2}···P_{n})}a_{P_{11}}a_{P_{22}}···a_{P_{nn}}

一、转置行列式：将D的行列互换（a_{ij} = b_{ji}）得到D^T 为D的转置行列式

D = \left|\begin{array}{ccc}a_{ij} \end{array}\right|=\left|\begin{array}{ccc}\color{red} {a_{11}} & \color{red} {a_{12}} & ··· & \color{red} {a_{1n}} \\\color{blue} {a_{21}} & \color{blue} {a_{22}} & ··· & \color{blue} {a_{2n}} \\··· & ··· & ··· & ··· \\\color{green} {a_{n1}} & \color{green} {a_{n2}} & ··· & \color{green} {a_{nn}} \end{array}\right|D^T =\left|\begin{array}{ccc}b_{ij} \end{array}\right|=\left|\begin{array}{ccc}\color{red} {a_{11}} & \color{blue} {a_{21}} & ··· & \color{green} {a_{n1}} \\\color{red} {a_{12}} & \color{blue} {a_{22}} & ··· & \color{green} {a_{n2}} \\··· & ··· & ··· & ··· \\\color{red} {a_{1n}} & \color{blue} {a_{2n}} & ··· & \color{green} {a_{nn}} \end{array}\right|

性质一、行列式与它的转置行列式相等，因此行列具有相同的性质，对行成立的性质对列同样成立，反之亦然

推导：

D =\left|\begin{array}{ccc}a_{ij} \end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···a_{nP_{n}} (行列式的行顺序定义)

D^T =\left|\begin{array}{ccc}b_{ij} \end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}b_{P_{1}1}b_{P_{2}2}···b_{P_{n}n} （行列式的列顺序定义）

由a_{ij} = b_{ji}\Rightarrow D=D^T

性质二、互换行列式的两行（或两列），行列式变号

推导：

D = \left|\begin{array}{ccc}a_{ij} \end{array}\right| =\left|\begin{array}{ccc}\color{} {a_{11}} & \color{} {a_{12}} & ··· & \color{} {a_{1n}} \\··· & ··· & ··· & ··· \\\color{red} {a_{x1}} & \color{red} {a_{x2}} & ··· & \color{red} {a_{xn}} \\··· & ··· & ··· & ··· \\\color{blue} {a_{y1}} & \color{blue} {a_{y2}} & ··· & \color{blue} {a_{yn}} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···\color{red}{P_{x}}···\color{blue}{P_{y}}···P_{n})}a_{1P_{1}}a_{2P_{2}}···\color{red}{a_{xP_{x}}}···\color{blue}{a_{yP_{y}}}···a_{nP_{n}}

D_{2} = \left|\begin{array}{ccc}a_{ij} \end{array}\right| =\left|\begin{array}{ccc}\color{} {a_{11}} & \color{} {a_{12}} & ··· & \color{} {a_{1n}} \\··· & ··· & ··· & ··· \\\color{blue} {a_{y1}} & \color{blue} {a_{y2}} & ··· & \color{blue} {a_{yn}} \\··· & ··· & ··· & ··· \\\color{red} {a_{x1}} & \color{red} {a_{x2}} & ··· & \color{red} {a_{xn}} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···\color{blue}{P_{y}}···\color{red}{P_{x}}···P_{n})}a_{1P_{1}}a_{2P_{2}}···\color{blue}{a_{yP_{y}}}···\color{red}{a_{xP_{x}}}···a_{nP_{n}}

每一项逆序数都对换1次（奇数次）即D_{2}相对D所有累加数符号都发生了变化 \Rightarrow D = -D_{2}

推论一：奇数次互换变号，偶数次互换不变

            推导：D=-D_{2} \ \&\&\ D_{2}=-D_{3}\Rightarrow D=D_{3}

推论二：如果行列式有两行（或两列）完全相同，则行列式等于0

            推导：D= -D\Rightarrow D=0

性质三、行列式的某一行（或一列）中所有元素同时乘以一个数k,等同于用数k乘以行列式

推导：

D = \left|\begin{array}{}a_{ij} \end{array}\right| =\left|\begin{array}{}\color{} {a_{11}} & \color{} {a_{12}} & ··· & \color{} {a_{1n}} \\··· & ··· & ··· & ··· \\\color{red} {k}a_{x1} & \color{red} {k}a_{x2} & ··· & \color{red} {k}a_{xn} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···\color{red}{k}a_{xP_{x}}···a_{nP_{n}} \\ \ \\=\color{red}{k}\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···a_{xP_{x}}···a_{nP_{n}}

推论一：行列式某一行（或一列）的公因子可以提到行列式外

推论二：行列式外的乘积影子可以乘到行列式某一行（或一列）中

性质四、如果行列式有两行（或两列）元素成比例，则行列式等于0

推导：

D = \left|\begin{array}{ccc}a_{ij} \end{array}\right| =\left|\begin{array}{ccc}\color{} {a_{11}} & \color{} {a_{12}} & ··· & \color{} {a_{1n}} \\··· & ··· & ··· & ··· \\\color{red}{k}\color{blue}{x_{1}} & \color{red}{k}\color{blue}{x_{2}} & ··· & \color{red}{k}\color{blue}{x_{n}} \\··· & ··· & ··· & ··· \\\color{blue}{x_{1}} & \color{blue} {x_{2}} & ··· & \color{blue} {x_{n}} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|=\color{red}{k}\left|\begin{array}{ccc}\color{} {a_{11}} & \color{} {a_{12}} & ··· & \color{} {a_{1n}} \\··· & ··· & ··· & ··· \\\color{blue}{x_{1}} & \color{blue}{x_{2}} & ··· & \color{blue}{x_{n}} \\··· & ··· & ··· & ··· \\\color{blue}{x_{1}} & \color{blue} {x_{2}} & ··· & \color{blue} {x_{n}} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|

由性质二-推论二可知 两行或两列完全相同，则行列式为0  \Rightarrow D = k*0 = 0

性质五、如果行列式某一行（或一列）中所有的元素都是两数之和，则可分解成两个行列式的和

推导：

D = \left|\begin{array}{}a_{ij} \end{array}\right| =\left|\begin{array}{}a_{11} & a_{12} & ··· & a_{1n} \\··· & ··· & ··· & ··· \\\color{red}{b_{x1}} + \color{blue}{c_{x1}} & \color{red}{b_{x2}} + \color{blue}{c_{x2}} & ··· & \color{red}{b_{xn}} + \color{blue}{c_{xn}} \\··· & ··· & ··· & ··· \\a_{n1} & a_{n2} & ··· & a_{nn} \\\end{array}\right|=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···(\color{red}{b_{xP_{x}}} + \color{blue}{c_{xP_{x}}})···a_{nP_{n}} \\ \ \\=\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···\color{red}{b_{xP_{x}}}···a_{nP_{n}}+\sum\nolimits_{1}^n(-1)^{ \tau(P_{1}P_{2}···P_{n})}a_{1P_{1}}a_{2P_{2}}···\color{red}{c_{xP_{x}}}···a_{nP_{n}}


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