Determinant and its related

Axiomatic definition of determinant

Generally speaking linear algebra, first of all, The matrix theory, and then the determinant, and then the linear transformation, linear space, eigenvalue theory, two-dimensional theory, and some domestic teaching materials such as the sixth edition of "linear algebra" first said the determinant of the matrix theory, simply anti-human, hehe. Recommended Use

-Professor Gilbert-strang's "Introduction to Linear Algebra" has now come out to the fifth edition, NetEase has strange professor's public lesson video.
-Professor Igor Shafarevich Linear Algebra and Geomerty
-Alex, "linear algebra should be learned"
-Blue to Middle "advanced algebra Concise Tutorial"

Now that the determinant is first, it is advisable to discard the concrete definition of the determinant, consider it as a function of some nature, and establish the determinant theory by axiomatic method.

define 1 (determinant) set $m_{n\times n}$ as a set of all components of the $n$ order matrix, map $f:m_{n\times n}\to \mathbb{r}$, and map $f$ satisfy

(P1) The determinant of the unit matrix is 1, which is $f (E_{1},e_{2},\cdots,e_{n}) =1.$

(P2) interchange matrix $a\in m_{n\times n}$ any two lines $\alpha_{j}$ and $\alpha_{k}$, determinant change symbols, i.e. $f (\alpha_{1},\alpha_{2},\cdots,\alpha_{j},\ Cdots,\alpha_{k},\cdots,\alpha_{n}) =-f (\alpha_{1},\alpha_{2},\cdots,\alpha_{k},\cdots,\alpha_{j},\cdots,\alpha _{n}) $.

(P3) mapping $f$ is linear to the $k$ variable, i.e.

$f (\alpha_{1},\alpha_{2},\cdots,\alpha_{k}+\beta_{k},\cdots,\alpha_{n}) =f (\alpha_{1},\alpha_{2},\cdots,\alpha_ {k},\cdots,\alpha_{n}) +f (\alpha_{1},\alpha_{2},\cdots,\beta_{k},\cdots,\alpha_{n}), $

$f (\alpha_{1},\alpha_{2},\cdots,t\alpha_{k},\cdots,\alpha_{n}) =t F (\alpha_{1},\alpha_{2},\cdots,\alpha_{k},\ Cdots,\alpha_{n}), \,\,t\in\mathbb{r}.$

If there is only one mapping that satisfies the nature P (1), P (2), P (3) * * $f: M_{n\times n}\to \mathbb{r}$, then $f (\alpha_{1},\alpha_{2},\cdots,\alpha_{k},\ Cdots,\alpha_{n}) $ is the determinant of the Matrix $a$, recorded as $\det (A) $ or $| a|$. Before we determine the uniqueness of this mapping, we still use $f (\alpha_{1},\alpha_{2},\cdots,\alpha_{k},\cdots,\alpha_{n}) $ or $f (A) $ notation, See if you can roll out more of the other properties of the determinant.

By the definition of determinant can be introduced by the nature of

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(P4) If the two rows of the square are the same, the determinant of the matrix is zero.

Proof. Exchange the position of the same two lines, depending on the nature P (2) has
$$
F (\alpha_{1},\cdots,\alpha_{k},\cdots,\alpha_{k},\cdots,\alpha_{n}) =-f (\alpha_{1},\cdots,\alpha_{k},\cdots,\ Alpha_{k},\cdots,\alpha_{n}),
$$
So $f (A) =0$.

P (5) $t$ a row of a matrix to another row, and the determinant does not change.

Proof. According to the Nature P (3) has $f (\alpha_{1},\cdots,\alpha_{j}+t\alpha_{k},\cdots,\alpha_{k},\cdots,\alpha_{n}) =f (\alpha_{1},\cdots, \alpha_{i},\cdots,\alpha_{k},\cdots,\alpha_{n}) +TF (\alpha_{1},\cdots,\alpha_{k},\cdots,\alpha_{k},\cdots,\ Alpha_{n}). $

According to the Nature P (4), $f (\alpha_{1},\cdots,\alpha_{k},\cdots,\alpha_{k},\cdots,\alpha_{n}) =0$.

So
$$
F (\alpha_{1},\cdots,\alpha_{j}+t\alpha_{k},\cdots,\alpha_{k},\cdots,\alpha_{n}) =f (\alpha_{1},\cdots,\alpha_{i} , \cdots,\alpha_{k},\cdots,\alpha_{n}).
$$
If a row of the P (6) matrix is all zeros, the determinant of the matrix is zero.

Proof. According to the Nature P (3), easy to get $f (a) =2f (a) $, so $f (a) =0.$

P (7) Upper triangular determinant

$$
U=\left (\BEGIN{ARRAY}{CCCC} a_{11} & A_{12} & \cdots & a_{1n} \ &a_{22} & \cdots &a_{2n} \ & & \ddots & \vdots \ \ {\HUGE0} & & &a_{nn}\end{array}\right)
$$

Determinant of $f (U) =a_{11}a_{22}\cdots a_{nn}.$

Proof. It is advisable to set $A_{JJ}\NEQ 0, j=1,2,\cdots,n.$ by the $gauss-jordan$ elimination method, * * * * * * * * * * with a constant of a row to the other row of the elementary row Edge line transformation
$$
U=\left (\BEGIN{ARRAY}{CCCC} a_{11} & A_{12} & \cdots & a_{1n} \ &a_{22} & \cdots &a_{2n} \ & & \ddots & \vdots \ {\huge0} & & &a_{nn}\end{array}\right) \to\widetilde{u}=\left (\begin{array}{ CCCC} a_{11} & & & \ &a_{22} & & \ & & \ddots & \ & & &a_{nn}\end{array }\right),
$$
According to P (5) there is, $f (U) =f (\widetilde{u}). $ And according to P (3) easy to get $f (\widetilde{u}) =a_{11}a_{22}\cdots a_{nn}.$

If $\exists j\in \{1,2,\cdots,n\},s.t. a_{jj}=0.$, the rank of matrix is not changed by the elementary line transformation,
$$
U=\left (\BEGIN{ARRAY}{CCCC} a_{11} & A_{12} & \cdots & a_{1n} \ &a_{22} & \cdots &a_{2n} \ & & \ddots & \vdots \ {\huge0} & & &a_{nn}\end{array}\right) \to\widetilde{u}=\left (\begin{array}{ CCCC} \widetilde{a}_{11} & & & \ &\widetilde{a}_{22} & & \ & & \ddots & \ & &am P &0\end{array}\right),
$$
Thus $f (U) =f (\widetilde{u}) =0=a_{11}a_{22}\cdots a_{nn}.$

Similarly, the determinant of the lower triangular matrix is $f (L) =a_{11}a_{22}\cdots a_{nn}.$

If there is a mapping $f:m_{n\times n}\to \mathbb{r}$ that satisfies the nature of satisfying nature P (1), P (2), P (3), then the inference must be unique.

Proof. Uniqueness of the equivalent standard type of matrix $a$ (the elementary transformation is obtained only by exchanging two lines of order and the constant of a row to another line)
$$
A\to\widetilde{a}=\left (\BEGIN{ARRAY}{CCCC} \widetilde{a}_{11} & & & \ &\widetilde{a}_{22} & & \ & & \ddots & \ & & &\widetilde{a}_{nn}\end{array}\right),
$$
$f (a) =g (a) =\widetilde{a}_{11}\widetilde{a}_{22}\cdots \widetilde{a}_{nn}.\forall a\in M_{n\times n}. $ So $f=g$ is the only proof.

P (8) When the $a$ is a singular matrix, the $f (a) =0$, and if the $a$ is a reversible matrix, then $f (a) \neq 0.$ is also established.

Proof. Proof of thinking for $a\to U \to d.$ reference P (7) of the inference.

P (9) Determinant of matrix multiplication
$$
F (AB) =f (A) f (B).
$$
Proof. It is advisable to set $a,\,b$ as non-singular matrices, otherwise
$$
Rank (AB) \leq \min\{rank (A), rank (B) \}<n,
$$
By the nature of P (8) to be
$$
F (AB) =f (A) F (B) =0.
$$
Set
$$
G (B) =\frac{f (AB)}{f (A)}
$$
It is easy to verify that $g (b) satisfies the nature of P (1), P (2), P (2) by the syllogism of P (7) and the uniqueness of the determinant is $f (b) =g (b) $, so the proposition is proved.

Inference $f (A^{m}) =f^{m} (A), \,\,m\geq-1,\,m\in \mathbb{z}.$

The determinant of the P (10) matrix is the same as the determinant of the transpose matrix.

Proof. The $lu$ decomposition of the matrix by the Gauss elimination method, that is $a=lu$, wherein the $l$ is the lower triangular matrix, $U $ for the upper triangular matrix. By the Nature P (9) and P (7) are
$$
F (A) =f (LU) =f (L) F (U) =f (L^{t}) F (U^{t}) =f (U^{t}l^{t}) =f (A^{t}).
$$

The existence of determinant function

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The basic idea is to constantly use linearity and P (1)-P (8) to get
$$
\det (A) =\sum_{p_{1}p_{2}\cdots P_{n}} ( -1) ^{\tau (p_{1}p_{2}\cdots p_{n})}a_{1\,p_{1}}a_{2\,p_{2}}\cdots A_{n\,p_{n }}.
$$
Note here that $p_{1}p_{2}\cdots p_{n}$ is an arrangement of $123\cdots n$ that does not represent the product, and $\tau (P_{1}p_{2}\cdots p_{n}) is the number of reverse order for that arrangement. In this way, the existence is obtained, and the inference of P (7) proves the existence and uniqueness of the determinant function.

A byproduct is
$$
\det (A) =a_{11}a_{11}+a_{22}a_{22}+\cdots+a_{nn}a_{nn},
$$
This formula can be reduced to a determinant, so a new determinant definition can be given using mathematical induction. Just define the first-order determinant!

Concrete calculation of determinant

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Method One: Make use of elementary row (or column) transform to become diagonal matrix. (commonly known as "punch hole" * *)

Method Two: Continuously descending the order.

Method Three: Other odd kinky tricks.

The application of determinant

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The applications that can be recalled and determinant related are

-Determine if the matrix is reversible
-Finding the eigenvalues of a matrix
-The Lex law of the linear equation group
-positive characterization of symmetric matrices with two-sub-types
-The geometrical meaning (area, volume) of second-order and third-rank determinant, and its application in plane and solid geometry
-The relationship between the Jocobi determinant and the variable substitution of the re-integral and the external differential
-Linear correlation of Wronsky determinant and function
-Existence of Vandermonde determinant and Lagrange interpolation polynomial

Determinant and its related