Symmetric matrices and normal matrices
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
Let $A$ be a complex matrix.
If $A^* = A$, then $A$ is Hermitian .
If $A^*A = AA^*$, then $A$ is normal .
Following the Schur triangulation theorem, these matrices can be diagonalized nicely.
Let $A$ be a normal matrix.
Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal.
Equivalently, $A$ has an orthonormal eigenbasis over $\mathbb{C}$.
Let $A$ be a Hermitian matrix.
Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal and real.
Equivalently, $A$ has an orthonormal eigenbasis over $\mathbb{C}$ and its eigenvalues are all real.
Let $A$ be a real matrix.
Then $A$ is normal if and only if $A\trans A = AA\trans$.
Thus, $A$ has an orthonormal basis over $\mathbb{C}$ (not necessarily over $\mathbb{R}$).
Similarly, $A$ is Hermitian if and only if $A\trans = A$.
That is, $A$ is a symmetric matrix.
The spectral theorem ensures that $A$ has an orthonormal basis over $\mathbb{C}$ and its all eigenvalues real.
It requires a few more steps to say the basis can actually be taken over $\mathbb{R}$.
Let $A$ be a real symmetric matrix.
Then there is a real orthogonal matrix $Q$ such that $Q\trans AQ$ is diagonal and real.
Equivalently, $A$ has an orthonormal eigenbasis over $\mathbb{R}$ and its eigenvalues are all real.
執行以下程式碼。
Run the code below.### code
set_random_seed(0)
print_ans = False
while True:
eigs = random_int_list(2)
v1 = vector(random_int_list(2))
if eigs[0] != eigs[1] and v1.is_zero() == False:
break
v2 = vector([-v1[1], v1[0]])
Q = matrix([v1.normalized(), v2.normalized()]).transpose()
A = Q * diagonal_matrix(eigs) * Q.inverse()
pretty_print(LatexExpr("A ="), A)
if print_ans:
print("eigenvalues of A:", eigs)
pretty_print(LatexExpr("Q ="), Q)
求 $A$ 的所有特徵值。
Find the spectrum of $A$.找一個實垂直矩陣 $Q$ 使得 $Q\trans AQ$ 為一對角矩陣。
Find a real orthogonal matrix $Q$ such that $Q\trans AQ$ is a diagonal matrix.將以下矩陣以實垂直矩陣對角化。
Diagonalize the following matrices by real orthogonal matrices.將以下矩陣以么正矩陣對角化。
Diagonalize the following matrices by unitary matrices.將
$$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $$以實垂直矩陣對角化。
Diagonalize $$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $$ by a real orthogonal matrix.將
$$ A = \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \end{bmatrix} $$以實垂直矩陣對角化。
Diagonalize $$ A = \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \end{bmatrix} $$ by a real orthogonal matrix.令 $T$ 為一上三角複矩陣。
Let $T$ be a complex upper triangular matrix.證明以下敘述等價:
證明以下敘述等價:
利用第 6 題及薛爾上三角化證明以下定理。
Use 6 and the Schur triangulation lemma to prove the following theorems.證明:
Let $A$ be a normal matrix.
Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal.
Equivalently, $A$ has an orthonormal basis.
Prove: ##### Spectral theorem (normal matrix) Let $A$ be a normal matrix. Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal. Equivalently, $A$ has an orthonormal basis.證明:
Let $A$ be a Hermitian matrix.
Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal and real.
Equivalently, $A$ has an orthonormal basis and its eigenvalues are all real.
Prove: ##### Spectral theorem (Hermitian matrix) Let $A$ be a Hermitian matrix. Then there is a unitary matrix $Q$ such that $Q^* AQ$ is diagonal and real. Equivalently, $A$ has an orthonormal basis and its eigenvalues are all real.證明:
Let $A$ be a real symmetric matrix.
Then there is a real orthogonal matrix $Q$ such that $Q\trans AQ$ is diagonal and real.
Equivalently, $A$ has an orthonormal basis over $\mathbb{R}$ and its eigenvalues are all real.
Prove: ##### Spectral theorem (symmetric matrix) Let $A$ be a real symmetric matrix. Then there is a real orthogonal matrix $Q$ such that $Q\trans AQ$ is diagonal and real. Equivalently, $A$ has an orthonormal basis over $\mathbb{R}$ and its eigenvalues are all real.實際上,要證明一個自伴矩陣的特徵值均為實數
不一定要用到譜定理。
令 $A$ 為一自伴矩陣。
若 $A\bx = \lambda \bx$,考慮 $\bx^* A \bx$ 及其共軛轉置。
藉此說明 $A$ 的特徵值均為實數。
以下練習探討么正矩陣和實垂直矩陣的相關性質。
The following exercises studies the basic properties of unitary matrices and real orthogonal matrices.說明么正矩陣是正規矩陣。
藉此說明么正矩陣可以被么正矩陣對角化。
並證明其特徵值的絕對值均為 $1$。
說明實垂直矩陣可以被么正矩陣對角化、
其特徵值的絕對值均為 $1$、
而且其特徵值對實軸對稱。