Algebraic multiplicity and geometric multiplicity
This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
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from lingeo import random_int_list, random_good_matrix
We have seen whether a matrix $A$ is diagonalizable depends on how many (independent) vectors we can find in $\ker(A - \lambda I)$ for each eigenvalue $\lambda\in\spec(A)$.
In this section, we will give some quantitative measures on how much a matrix is diagonalizable.
Let $A$ be an $n\times n$ matrix and $p_A(x)$ its characteristic polynomial.
For each $\lambda \in \spec(A)$,
When the context is clear, the subscript $A$ can be omitted.
For any $n\times n$ matrix $A$ and $\lambda\in\spec(A)$, the following properties hold.
The fact $1\leq\gm(\lambda)$ follows immediate from the definition that $A - \lambda I$ is singular when $\lambda$ is an eigenvalue.
The exercises contains a proof of $\gm(\lambda) \leq \am(\lambda)$.
The sum of $\am(\lambda)$ is equal to the degree of $p_A(x)$, which is $n$.
If $\gm(\lambda) < \am(\lambda)$ for some $\lambda$, then the sum of all geometric multiplicities is less than $n$, so it is impossible to find a basis composed of eigenvectors.
If $\gm(\lambda) = \am(\lambda)$ for all distinct eigenvalues $\lambda$, we will show in the next section that $A$ is diagonalizable.
Let $A$ be an $n\times n$ matrix.
If the characteristic polynomial $p_A(x)$ has $n$ distinct roots, then $A$ is diagonalizable.
### code
set_random_seed(0)
print_ans = False
am1 = choice([2,3])
am2 = choice([2,3])
n = am1 + am2
while True:
lam1, lam2 = random_int_list(2,3)
if lam1 != lam2:
break
gm1 = randint(1,am1)
gm2 = randint(1,am2)
block1 = [matrix([[lam1]]) for i in range(gm1 - 1)] + [jordan_block(lam1, am1 - gm1 + 1)]
block2 = [matrix([[lam2]]) for i in range(gm2 - 1)] + [jordan_block(lam2, am2 - gm2 + 1)]
D = block_diagonal_matrix(block1 + block2)
Q = random_good_matrix(n,n,n,2)
A = Q * D * Q.inverse()
pretty_print(LatexExpr("A ="), A)
pA = (-1)^n * A.charpoly()
print("characteristic polyomial =", pA)
print(" =", factor(pA))
print("rref of A - (%s)I:"%lam1)
pretty_print((A - lam1).rref())
print("rref of A - (%s)I:"%lam2)
pretty_print((A - lam2).rref())
if print_ans:
print("lambda =", lam1)
print("am(lambda) =", am1)
print("gm(lambda) =", gm1)
print("lambda =", lam2)
print("am(lambda) =", am2)
print("gm(lambda) =", gm2)
print("Diagonalizable?", am1 == gm1 and am2 == gm2)
求 $A$ 的每一個特徵值的代數重數與幾何重數。
For each eigenvalue of $A$, find its algebraic and geometric multiplicities.
對以下矩陣,
計算每一個特徵值的代數重數與幾何重數,
並判斷其是否可以對角化。
For each of the following matrices, find the algebraic and geometric multiplicities of its eigenvalues, and determine if it is diagonalizable.
對以下矩陣,
計算每一個特徵值的代數重數與幾何重數,
並判斷其是否可以對角化。
For each of the following matrices, find the algebraic and geometric multiplicities of its eigenvalues, and determine if it is diagonalizable.
以下矩陣稱為喬丹區塊矩陣(Jordan block)。
說明以下矩陣
不計算重數的話只有一個特徵值,
計算這個特徵值的代數重數與幾何重數。
因此大小大於等於 $2$ 的喬丹區塊矩陣皆不可對角化。
The following matrices are examples of Jordan blocks. Show that each of the following matrices only has one distinct eigenvalue. Find the algebraic and geometric multiplicities of this eigenvalue.
找尋滿足以下條件的矩陣。
For each of the following exercises, find a matrix that meet the conditions.
找一個矩陣 $A$ 使得
$\spec(A) = \{2,2,3,3,3\}$、
$\gm(2) = 1$ 且 $\gm(3) = 2$。
Find a matrix $A$ such that $\spec(A) = \{2,2,3,3,3\}$, $\gm(2) = 1$, and $\gm(3) = 2$.
找一個每一項都不為零的矩陣 $A$ 使得
$\spec(A) = \{2,2,3,3,3\}$、
$\gm(2) = 1$ 且 $\gm(3) = 2$。
Find a matrix $A$ whose entries are all nonzero such that $\spec(A) = \{2,2,3,3,3\}$, $\gm(2) = 1$, and $\gm(3) = 2$.
令 $A$ 為一 $n\times n$ 矩陣而 $\lambda$ 為其一特徵值。
依照以下步驟證明 $\gm(\lambda) \leq \am(\lambda)$。
Let $A$ be an $n\times n$ matrix and $\lambda$ one of its eigenvalues. Use the given instructions to show that $\gm(\lambda) \leq \am(\lambda)$.
令 $d = \gm(\lambda)$ 為 $\ker(A - \lambda I)$ 的維度。
找一組 $\ker(A - \lambda I)$ 的基底 $\alpha$,
並將其擴展為 $\mathbb{R}^n$ 的一組基底 $\beta$。
(所以 $\alpha\subseteq\beta$ 且 $|\alpha| = d$。)
說明 $[f_A]_\beta^\beta$ 會有以下型式
(這裡 $D$ 只是一個矩陣的符號,不一定是對角矩陣。)
Let $d = \gm(\lambda)$ be the dimension of $\ker(A - \lambda I)$. Let $\alpha$ be a basis of $\ker(A - \lambda I)$. Then extend it into a basis $\beta$ of $\mathbb{R}^n$. (Therefore, $\alpha\subseteq\beta$ and $|\alpha| = d$.) Show that $[f_A]_\beta^\beta$ has the form
$$ \begin{bmatrix} \lambda I_d & B \\ O_{n-d,d} & D \end{bmatrix}. $$(Here $D$ is just a matrix, which is not necessarily diagonal.)
由於 $[f_A]_\beta^\beta$ 和 $A$ 有相同的特徵多項式。
說明 $p_A(x)$ 中有 $(\lambda - x)^d$ 這個因式,
因此 $\gm(\lambda) \leq \am(\lambda)$。
Observe that $[f_A]_\beta^\beta$ and $A$ have the same characteristic polynomial. Show that $(\lambda - x)^d$ is a factor of $p_A(x)$. Therefore, $\gm(\lambda) \leq \am(\lambda)$.
令 $A$ 為一 $n\times n$ 矩陣而 $W$ 為一 $\mathbb{R}^n$ 的子空間。
如果
則我們稱 $W$ 是一個 $A$-不變子空間($A$-invariant subspace)。
同樣地,令 $f:V\rightarrow V$ 為一個線性函數
而 $W$ 是 $V$ 的一個子空間。
如果
則我們稱 $W$ 是一個 $f$-不變子空間($f$-invariant subspace)。
Let $A$ be an $n\times n$ matrix and $W$ a subspace of $\mathbb{R}^n$. If
$$ f_A(W) = \{A\bw: \bw\in W\} \subseteq W, $$then $W$ is called an $A$-invariant subspace .
Similarly, let $f:V\rightarrow V$ be a linear function and $W$ a subspace of $V$. If
$$ f(W) = \{f(\bw): \bw\in W\} \subseteq W, $$then $W$ is called a $f$-invariant subspace .
令 $A$ 為一 $n\times n$ 矩陣而 $W$ 為一 $A$-不變子空間。
令 $\alpha$ 為 $W$ 的一組基底
並將基擴展為 $\mathbb{R}^n$ 的一組基底 $\beta$。
說明 $[f_A]_\beta^\beta$ 會有以下型式
(這裡 $D$ 只是一個矩陣的符號,不一定是對角矩陣。)
Let $A$ be an $n\times n$ matrix and $W$ an $A$-invariant subspace. Let $\alpha$ be a basis of $W$. Then extend it into a basis $\beta$ of $\mathbb{R}^n$. Show that $[f_A]_\beta^\beta$ has the form
$$ \begin{bmatrix} \lambda A' & B \\ O_{n-d,d} & D \end{bmatrix}. $$(Here $D$ is just a matrix, which is not necessarily diagonal.)
若 $A$ 為一矩陣,而 $\lambda$ 為其一特徵值。
說明 $\ker(A - \lambda I)$ 為一 $A$-不變子空間。
Let $A$ be an $n\times n$ matrix and $\lambda$ one of its eigenvalues. Show that $\ker(A - \lambda I)$ is an $A$-invariant subspace.