# 體驗喬丹標準型

Understand the Jordan canonical form

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, random_good_matrix

## Main idea

Let $\lambda\in\mathbb{C}$ and $n$ an integer.
A **Jordan block** of $\lambda$ of order $n$ is the $n\times n$ matrix

$$
J_{\lambda,n} = \begin{bmatrix}
\lambda & 1 & & \\
& \ddots & \ddots & \\
& & & 1 \\
& & & \lambda \\
\end{bmatrix}.
$$

According to the spectrum theorem, for every symmetric matrix $A$, there is an orthonormal basis $\beta$ such that $[f_A]_\beta^\beta$ is orthogonal.
This is not necessarily true for not necessarily symmetric matrices, even we give up the condition of being orthonormal.
Any Jordan block of order at least $2$ is such an example.
Fortunately, we have the folloing canonical form.

##### Jordan canonical form

Let $A$ be an $n\times n$ matrix.
Then there is a basis $\beta$ of $\mathbb{R}^n$ such that

$$
[f_A]_\beta^\beta = J_{\lambda_1,n_1}\oplus \cdots \oplus J_{\lambda_d,n_d},
$$

where $\oplus$ is the direct sum of two matrices.

That is, there is an invertible matrix $Q$ such that $Q^{-1}AQ = J_{\lambda_1,n_1}\oplus \cdots \oplus J_{\lambda_d,n_d}$.

We say two $n\times n$ matrices $A$ and $B$ are **similar** if there is an invertible matrix $Q$ such that $Q^{-1}AQ = B$.
Equivalently, $A$ and $B$ are similar if there is a basis $\beta$ of $\mathbb{R}^n$ such that $[f_A]_\beta^\beta = B$.
It is not trivial how to determine whether two given matrices are similar.
It turns out that two matrices are similar if and only if they have the same Jordan canonical form.

## Side stories

- matrix algebra

## Experiments

##### Exercise 1

執行以下程式碼。
已知 $A$ 和 $B$ 相似﹐且 $Q^{-1}AQ = B$。

<!-- eng start -->
Run the code below. Suppose $A$ and $B$ are similar and $Q^{-1}AQ = B$.
<!-- eng end -->

### code
set_random_seed(0)
print_ans = False
n = 2
A = matrix(2, random_int_list(n*n))
Q = random_good_matrix(n,n,n)
B = Q.inverse() * A * Q

print("Qinv * A * Q = B")
pretty_print(Q.inverse(), A, Q, "=", B)

if print_ans:
    print("rank =", A.rank())
    print("nullity =", A.nullity())
    print("determinant =", A.determinant())
    print("B (Qinv v) = (Qinv b)")

##### Exercise 1(a)

驗證 $A$ 和 $B$ 有相同的秩、核數、行列式值。

<!-- eng start -->
Verify that $A$ and $B$ have the same rank, nullity, and determinant.
<!-- eng end -->

##### Exercise 1(b)

若已知 $A\bv = \bb$。
求 $B(Q^{-1}\bv)$。

<!-- eng start -->
Suppose $A\bv = \bb$. Then find $B(Q^{-1}\bv)$.
<!-- eng end -->

## Exercises

##### Exercise 2

令 $A$ 和 $B$ 為兩 $n\times n$ 矩陣。
證明以下敘述等價：

1. $A$ 和 $B$ 相似。
2. 存在 $\mathbb{R}^n$ 的一組基底 $\beta$ 使得 $[f_A]_\beta^\beta = B$。

<!-- eng start -->
Let $A$ and $B$ be $n\times n$ matrices. Show that the following are equivalent:

1. $A$ and $B$ are similar.
2. There is a basis $\beta$ of $\mathbb{R}^n$ such that $[f_A]_\beta^\beta = B$.
<!-- eng end -->

##### Exercise 3

令

$$
J = J_{2,3} =
\begin{bmatrix}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2 \\
\end{bmatrix}.
$$

<!-- eng start -->
Let

$$
J = J_{2,3} =
\begin{bmatrix}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2 \\
\end{bmatrix}.
$$
<!-- eng end -->

##### Exercise 3(a)

說明若存在一組基底 $\beta = \{ \bv_1,\ldots,\bv_3 \}$ 使得 $[f_J]_\beta^\beta$ 是對角矩陣 $\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)$﹐
則對每個 $i = 1,\ldots, 3$ 都有 $J\bv_i = \lambda_i \bv_i$。

<!-- eng start -->
Show that if there is a basis $\beta = \{ \bv_1,\ldots,\bv_3 \}$ such that $[f_J]_\beta^\beta$ is a diagonal matrix $\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3)$, then $J\bv_i = \lambda_i \bv_i$ for each $i = 1, \ldots, 3$.
<!-- eng end -->

##### Exercise 3(b)

說明若這樣的基底存在﹐必定是 $\lambda_1 = \cdots = \lambda_3 = 2$﹐
然而 $J$ 不可能和 $2I$ 相似。

<!-- eng start -->
Show that if such a basis exists, then it must be $\lambda_1 = \cdots = \lambda_3 = 2$. However, $J$ is impossible to be similar with $2I$.
<!-- eng end -->

##### Exercise 4

令 $A$ 為一 $n\times n$ 矩陣。
已知 $\beta$ 為 $\mathbb{R}^n$ 的一組基底使得 $[f_A]_\beta^\beta = J = J_{\lambda,n}$。
將每個 $A\bv_i$ 寫成 $\beta$ 的線性組合。

<!-- eng start -->
Let $A$ be an $n\times n$ matrix. Suppose $\beta$ is a basis of $\mathbb{R}^n$ such that $[f_A]_\beta^\beta = J = J_{\lambda,n}$. Write each $A\bv_i$ as a linear combination of $\beta$.
<!-- eng end -->

##### Exercise 5

令

$$
A = \begin{bmatrix}
5 & 1 & 0 \\
-7 & -1 & 1 \\
-6 & -2 & 2
\end{bmatrix}.
$$

令
$\bv_1 = (1, -3, -2)$、
$\bv_2 = (1, -2, -2)$、
$\bv_3 = (2, -5, -3)$、
$\beta = \{ \bv_1, \ldots, \bv_3 \}$。
求 $[f_A]_\beta^\beta$。

<!-- eng start -->
Let

$$
A = \begin{bmatrix}
5 & 1 & 0 \\
-7 & -1 & 1 \\
-6 & -2 & 2
\end{bmatrix}.
$$

Let $\bv_1 = (1, -3, -2)$, $\bv_2 = (1, -2, -2)$, $\bv_3 = (2, -5, -3)$, and $\beta = \{ \bv_1, \ldots, \bv_3 \}$. Find $[f_A]_\beta^\beta$.
<!-- eng end -->

##### Exercise 6

證明兩矩陣相似是等價關係。

<!-- eng start -->
Show that the similarity between matrices is an equivalence relation.
<!-- eng end -->