# 行列式值是定義完善的

This work by Jephian Lin is licensed under a [Creative Commons Attribution 4.0 International License](http://creativecommons.org/licenses/by/4.0/).

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from lingeo import random_int_list
from gnm import effective_permutations, illustrate_det

## Main idea

In this section, we will review our treatment on the definition and the properties of the determinant function.

In 405, we defined the determinants of elementary matrices and define
$$
\det(A) = \det(F_1) \cdots \det(F_k)
$$
if $A = F_1\cdots F_k$ for some elementary matrices,
and $\det(A) = 0$ if $A$ is not invertible.

This help us to find the value of $\det(A)$
but it is not certain whether this definition is well-defined since $A$ can be written as different products of elementary matrices.

Based on the definition, we realized that the determinant, if well-defined, must satisfies the following properties:

- $\det(A\trans) = \det(A)$.
- $\det(AB) = \det(A)\det(B)$.
- distributive law on any row and any column.
- Laplace expansion along any row and any column.

Thanks to the Laplace expansion, we noticed that $\det(A)$, again, if well-defined, must equal to the value
$$
\sum_{\sigma\in\mathfrak{S}_n}\sgn(\sigma) w(\sigma).
$$
We tentatively call this formula of the permutation expansion as $\det_p(A)$.

If we focus only on the definition of $\det_p(A)$, then it is obviously well-defined.
And it is known if $\det(A)$ is well-defined, meaning there is no internal violation of the four rules in its definition, then $\det(A) = \det_p(A)$.
Therefore, all we need to do is to make sure $\det_p(A)$ satisfies the four rules in the definition of $\det(A)$.

## Side stories

- summary

## Experiments

##### Exercise 1

執行以下程式碼。

### code
set_random_seed(0)
print_ans = False

n = 6
b = 20
while True:
    l = random_int_list(n^2 - b, 3) + [0]*b
    shuffle(l)
    A = matrix(n, l)
    perms = effective_permutations(A)
    if len(perms) >= 3 and len(perms) <= 8 and A.det() != 0:
        break

B = copy(A)
B.swap_rows(0,1)
pretty_print(LatexExpr("A ="), A)
pretty_print(LatexExpr("B ="), B)


if print_ans:
    print("elementary subgraphs of Gamma(A):")
    illustrate_det(A)
    print("elementary subgraphs of Gamma(B):")
    illustrate_det(B)
    print("Take one elementary subgraph H_A for A.")
    print("If (1,i) and (2,j) are directed edges in H_A,")
    print("then changing them to (1,j) and (2,i) will result in an elementary subgraph H_B for B.")
    print("This builds up a bijection between the elementary subgraphs of Gamma(A) and Gamma(B).")

##### Exercise 1(a)

畫出 $\Gamma(A)$ 的所有基本子圖，並標上每條邊的權重。

##### Exercise 1(b)

畫出 $\Gamma(B)$ 的所有基本子圖，並標上每條邊的權重。

##### Exercise 1(c)

找一個 $\mathfrak{E}(\Gamma(A))$ 到 $\mathfrak{E}(\Gamma(B))$ 的一對一對應。
這個對應會把 $\mathfrak{E}(\Gamma(A))$ 對到 $\mathfrak{E}(\Gamma(B))$ 的一個權重一樣的基本子圖，
但是圈數差一，因此正負號會變號。

## Exercises

##### Exercise 2

找出 $\Gamma(I_n)$ 中的基本子圖，
並說明 $\det_p(I_n) = 1$。

##### Exercise 3

令 $A$ 是一矩陣，
而 $B$ 是由 $A$ 矩陣將第 $i$ 列和第 $j$ 列交換得來，且 $i \neq j$。
令 $\mathfrak{E}_A = \mathfrak{E}(\Gamma(A))$ 且 $\mathfrak{E}_B = \mathfrak{E}(\Gamma(B))$。
找一個對射函數 $f: \mathfrak{E}_A \rightarrow \mathfrak{E}_B$ 使得

- $w(f(H)) = w(H)$、
- $|c(f(H)) - c(H)|$、且
- $\sgn(f(H)) = - \sgn(H)$。

因此 $\det_p(B) = -\det_p(A)$。

##### Exercise 4

令 $A$ 是一矩陣，
而 $B$ 是由 $A$ 矩陣將第 $i$ 列乘上 $k$ 倍得來。
令 $\mathfrak{E}_A = \mathfrak{E}(\Gamma(A))$ 且 $\mathfrak{E}_B = \mathfrak{E}(\Gamma(B))$。
找一個對射函數 $f: \mathfrak{E}_A \rightarrow \mathfrak{E}_B$ 使得

- $w(f(H)) = k\cdot w(H)$、
- $c(f(H)) = c(H)$、且
- $\sgn(f(H)) = \sgn(H)$。

因此 $\det_p(B) = k\cdot\det_p(A)$。

##### Exercise 5

藉由前兩題，說明以下性質。

##### Exercise 5(a)

若 $A$ 中有兩列相同，則 $\det_p(A) = 0$。

##### Exercise 5(b)

若 $A$ 中有一列是另一列的倍數，則 $\det_p(A) = 0$。

##### Exercise 6

令 $M$ 是一矩陣，且其第一列可以寫成兩向量相加 $\ba + \bb$。
將 $M$ 的第一列改為 $\ba$ 並稱之為矩陣 $A$。
將 $M$ 的第一列改為 $\bb$ 並稱之為矩陣 $B$。
令 $\mathfrak{E}_M = \mathfrak{E}(\Gamma(M))$、$\mathfrak{E}_A = \mathfrak{E}(\Gamma(A))$ 且 $\mathfrak{E}_B = \mathfrak{E}(\Gamma(B))$。
找兩個對射函數 $f_A: \mathfrak{E}_M \rightarrow \mathfrak{E}_A$ 及 $f_B: \mathfrak{E}_M \rightarrow \mathfrak{E}_B$ 使得

- $w(f_A(H)) + w(f_B(H)) = w(H)$、
- $c(f_A(H)) = c(H)$ 及 $c(f_B(H)) = c(H)$、且
- $\sgn(f(H)) = \sgn(H)$ 及 $\sgn(f(H)) = \sgn(H)$。

因此 $\det_p(M) = \det_p(A) + \det_p(B)$。

##### Exercise 7

令 $A$ 是一矩陣，
而 $B$ 是由 $A$ 矩陣將第 $j$ 列乘上 $k$ 倍後加到第 $i$ 列得來，且 $i \neq j$。
利用前兩題的性質說明 $\det_p(B) = \det_p(A)$。