# 伴隨矩陣

Adjugate

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

## Main idea

Let $A$ be an $n\times n$ matrix.
The **$ij$-minor** of $A$ is $\det A(i,j)$, while
the **$ij$-cofactor** of $A$ is $(-1)^{i + j} \det A(i,j)$.
The **cofactor matrix** of $A$ is an $n\times n$ matrix $A\cof$ whose $ij$-entry is the $ij$-cofactor of $A$.

Let $\br_1,\ldots,\br_n$ be the rows of $A$.
Let $\bc_1,\ldots,\bc_n$ be the rows of $A\cof$.
Thus, the Laplace expansion can be written as

$$
\inp{\br_i}{\bc_i} = \det(A)
$$

for any $i = 1,\ldots, n$.

Suppose $i$ and $j$ are two different indices in $\{1,\ldots,n\}$.
Consider the matrix $B$ obtained from $A$ by replacing its $j$-th row with $\br_i$.
Since $B$ has repeated rows, $\det(B) = 0$.
Meanwhile, the $jk$-cofactor of $B$ is the same as the $jk$-cofactor of $A$ for any $k = 1,\ldots, n$.
This means

$$
\inp{\br_i}{\bc_j} = \det(B) = 0.
$$

In summary,

$$
\inp{\br_i}{\bc_j} =
\begin{cases}
\det(A) & \text{if }i = j, \\
0 & \text{if }i \neq j.
\end{cases}
$$

Define the **adjugate** of $A$ as $A\adj = (A\cof)\trans$.
Thus, the above summary leads to the identity

$$
AA\adj = \det(A)I = A\adj A.
$$

Therefore, when $\det(A) \neq 0$,

$$
A^{-1} = \frac{1}{\det(A)}A\adj.
$$

## Side stories

- unimodular
- outer product
- adjugate when $\rank(A) = n-1$ or $\rank(A) = n-2$
- Stirling numbers

## Experiments

##### Exercise 1

執行以下程式碼。

<!-- eng start -->
Run the code below.
<!-- eng end -->

### code
set_random_seed(0)
print_ans = False

n = 3
A = matrix(n, random_int_list(n^2,3))

pretty_print(LatexExpr("A ="), A) 

if print_ans:
    for i in range(n):
        for j in range(n):
            alpha = list(range(n))
            alpha.remove(i)
            beta = list(range(n))
            beta.remove(j)
            print("det A(%s,%s) ="%(i,j), A[alpha,beta].det())
    print("cofactor matrix:")
    pretty_print(A.adjugate().transpose())
    print("adjugate:")
    pretty_print(A.adjugate())

##### Exercise 1(a)

對所有 $i,j$，求出 $\det A(i,j)$。

<!-- eng start -->
For each $i,j$, find $\det A(i,j)$.
<!-- eng end -->

##### Exercise 1(b)

求 $A$ 的餘因子矩陣 $A\cof$。

<!-- eng start -->
Find the cofactor matrix $A\cof$ of $A$.
<!-- eng end -->

##### Exercise 1(c)

求 $A$ 的伴隨矩陣 $A\adj$。

<!-- eng start -->
Find the adjugate $A\adj$ of $A$.
<!-- eng end -->

:::info
What do the experiments try to tell you? (open answer)

...
:::

## Exercises

##### Exercise 2

根據拉普拉斯展開，計算行列式值時只會用到加法和乘法。
所以一個整數矩陣的行列式值也會是整數、
而一個有理數矩陣的行列式值也會是有理數。
利用這個性質回答以下問題。

<!-- eng start -->
According to the Laplace expansion, the computation of a determinant only uses the addition and the multiplication. Therefore, the determinant of an integer matrix is also an integer, and the determinant of a rational matrix is also a rational number. Use these facts to answer the following problems.
<!-- eng end -->

##### Exercise 2(a)

說明一個有理數矩陣（若可逆）的反矩陣也會是有理數矩陣。

<!-- eng start -->
Explain why the inverse of an invertible rational matrix is also a rational matrix.
<!-- eng end -->

##### Exercise 2(b)

找一個可逆的整數矩陣，其反矩陣不是整數矩陣。

<!-- eng start -->
Find an invertible integer matrix such that its inverse it not an integer matrix.
<!-- eng end -->

##### Exercise 2(c)

一個整數方陣如果行列式值為 $\pm 1$，
則被稱為**么模矩陣（unimodular matrix）** 。
說明么模矩陣的反矩陣一定是整數矩陣。

<!-- eng start -->
An integer matrix with determinant $\pm 1$ is called a **unimodular matrix** . Explain why the inverse of a unimodular matrix is also an integer matrix.
<!-- eng end -->

##### Exercise 3

令 $\br_1,\br_2,\br_3$ 為 $\mathbb{R}^3$ 中的向量、
且

$$
A = \begin{bmatrix}
- & \br_1 & - \\
~ & \vdots & ~ \\
- & \br_3 & -
\end{bmatrix}.
$$

說明 $A\cof$ 的第一列即為 $\br_2$ 和 $\br_3$ 的外積。
（根據定義，$A\cof$ 的第一列和 $\br_1$ 無關。）

<!-- eng start -->
Let $\br_1,\br_2,\br_3$ be vectors in $\mathbb{R}^3$ and

$$
A = \begin{bmatrix}
- & \br_1 & - \\
~ & \vdots & ~ \\
- & \br_3 & -
\end{bmatrix}.
$$

Explain why the first row of $A\cof$ is exactly the cross product of $\br_2$ and $\br_3$. (By definition, the first row of $A\cof$ is independent of $\br_1$.)
<!-- eng end -->

##### Exercise 4

給定 $\beta = \{\br_2,\ldots,\br_n\}$ 為 $\mathbb{R}^n$ 中的一群線性獨立的向量。
說明如何利用 $A\cof$ 找到一根向量 $\bv$ 使得 $\bv$ 跟 $\beta$ 中的所有向量都垂直。
（這個動作可以看成是 $\beta$ 中向量的外積。）

<!-- eng start -->
Let $\beta = \{\br_2,\ldots,\br_n\}$ be a linearly independent set in $\mathbb{R}^n$. Explain how to use $A\cof$ to find a vector $\bv$ such that $\bv$ is orthogonal to each vector in $\beta$. (You may also view this operation as the cross product of $\beta$.)
<!-- eng end -->

##### Exercise 5

令 $A$ 為一 $n\times n$ 矩陣。
當 $\rank(A) = n$ 時，我們知道 $A\adj = \det(A)A^{-1}$。

<!-- eng start -->
Let $A$ be an $n\times n$ matrix. When $\rank(A) = n$, we know $A\adj = \det(A)A^{-1}$.
<!-- eng end -->

##### Exercise 5(a)

當 $\rank(A) = n - 1$ 時，
可以假設 $A$ 的左右核分別為 $\ker(A) = \vspan\{\bu\}$ 及 $\ker(A\trans) = \vspan\{\bv\}$。
說明存在某個係數 $c$ 使得 $A\adj = c\bu\bv\trans$。

<!-- eng start -->
Suppose $\rank(A) = n - 1$. We may assume the left kernel and the right kernel of $A$ are $\ker(A) = \vspan\{\bu\}$ 及 $\ker(A\trans) = \vspan\{\bv\}$, respectively. Show that there is a constanct $c$ such that $A\adj = c\bu\bv\trans$.
<!-- eng end -->

##### Exercise 5(b)

當 $\rank(A) \leq n - 2$ 時，
說明 $A\adj = O$。
（提示：子矩陣的秩一定要比原矩陣的秩來得小。）

<!-- eng start -->
Suppose $\rank(A) = n - 2$. Show that $A\adj = O$. (Hint: The rank of a submatrix is smaller than the rank of the matrix itself.)
<!-- eng end -->

##### Exercise 6

定義 $(x)_k = x(x-1)\cdots(x-k+1)$ 且 $(x)_0 = 1$。
已知
$\alpha = \{1,x,\ldots, x_d\}$ 及
$\beta = \{(x)_0, (x)_1, \ldots, (x)_d\}$
皆是 $\mathcal{P}_d$ 的基底。

<!-- eng start -->
Define $(x)_k = x(x-1)\cdots(x-k+1)$ and $(x)_0 = 1$. Suppose we know $\alpha = \{1,x,\ldots, x_d\}$ and $\beta = \{(x)_0, (x)_1, \ldots, (x)_d\}$ are both bases of $\mathcal{P}_d$.
<!-- eng end -->

##### Exercise 6(a)

當 $d = 3$ 時，求出基底轉換矩陣 $[\operatorname{id}]_\beta^\alpha$。
並說明對任意的 $d$ 來說，$[\operatorname{id}]_\beta^\alpha$ 都會是整數矩陣
且行列式值為 $1$。

<!-- eng start -->
When $d = 3$, find the change-of-bases matrix $[\operatorname{id}]_\beta^\alpha$. Explain why $[\operatorname{id}]_\beta^\alpha$ is always an integer matrix with determinant $1$ for any $d$.
<!-- eng end -->

##### Exercise 6(b)

當 $d = 3$ 時，求出基底轉換矩陣 $[\operatorname{id}]_\alpha^\beta$。
利用上一題的結果，說明對任意的 $d$ 來說，$[\operatorname{id}]_\alpha^\beta$ 都會是整數矩陣。

<!-- eng start -->
When $d = 3$, find the change-of-bases matrix $[\operatorname{id}]_\alpha^\beta$. Explain why $[\operatorname{id}]_\beta^\alpha$ is always an integer matrix for any $d$.
<!-- eng end -->