Schur triangulation
This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
$\newcommand{\trans}{^\top} \newcommand{\adj}{^{\rm adj}} \newcommand{\cof}{^{\rm cof}} \newcommand{\inp}[2]{\left\langle#1,#2\right\rangle} \newcommand{\dunion}{\mathbin{\dot\cup}} \newcommand{\bzero}{\mathbf{0}} \newcommand{\bone}{\mathbf{1}} \newcommand{\ba}{\mathbf{a}} \newcommand{\bb}{\mathbf{b}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bh}{\mathbf{h}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bq}{\mathbf{q}} \newcommand{\br}{\mathbf{r}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bw}{\mathbf{w}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\nul}{\operatorname{null}} \newcommand{\rank}{\operatorname{rank}} %\newcommand{\ker}{\operatorname{ker}} \newcommand{\range}{\operatorname{range}} \newcommand{\Col}{\operatorname{Col}} \newcommand{\Row}{\operatorname{Row}} \newcommand{\spec}{\operatorname{spec}} \newcommand{\vspan}{\operatorname{span}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\idmap}{\operatorname{id}} \newcommand{\am}{\operatorname{am}} \newcommand{\gm}{\operatorname{gm}} \newcommand{\mult}{\operatorname{mult}} \newcommand{\iner}{\operatorname{iner}}$
from lingeo import random_int_list, random_good_matrix
Recall that an $n\times n$ matrix $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ is called upper triangular if $a_{ij} = 0$ for all $i > j$.
Let $A$ be a square complex matrix.
Then there is a unitary matrix $Q$ such that $Q^*AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
Note that if $A$ is a real matrix with complex eigenvalues, then the Schur triangulation theorem still work since $A$ can be viewed as a complex matrix.
However, the decomposition will enforce $Q$ and $T$ to have non-real entries.
Let $A$ be a square real matrix with all eigenvalues real.
Then there is a real orthogonal matrix $Q$ such that $Q\trans AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
Let $A$ be a square real matrix.
Then there is a real invertible matrix $Q$ such that $Q^{-1}AQ = T$ has the form
such that $B_k = \begin{bmatrix} \lambda_k \end{bmatrix}$ if $\lambda_k\in\mathbb{R}$
and $B_k = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ if $\lambda_k = a + bi$ with $b \neq 0$.
Necessarily, the diagonal blocks of $T$ determine the eigenvalues of $A$.
### code
set_random_seed(0)
print_ans = False
n = 4
eigs = random_int_list(n)
Q = random_good_matrix(n - 1, n - 1, n - 1, 3)
A2 = Q * diagonal_matrix(eigs[1:]) * Q.inverse()
Q2 = identity_matrix(n - 1)
Q2[:,0] = Q[:,0]
T2 = Q2.inverse() * A2 * Q2
A = zero_matrix(n, n)
A[0,0] = eigs[0]
A[0,1:] = vector(random_int_list(n - 1))
A[1:,1:] = A2
pretty_print(LatexExpr("Q_2^{-1} A_2 Q_2 ="), Q2.inverse(), A2, Q2, LatexExpr("="), T2)
pretty_print(LatexExpr("A ="), A)
if print_ans:
Qhat2 = block_diagonal_matrix(matrix([[1]]), Q2)
T = Qhat2.inverse() * A * Qhat2
pretty_print(LatexExpr(r"\hat{Q}_2^{-1} A_2 \hat{Q}_2 ="), Qhat2.inverse(), A, Qhat2, LatexExpr("="), T)
print("eigenvalues of A:", eigs)
令 $\hat{Q}_2 = 1 \oplus Q_2$。
求 $\hat{Q}_2^{-1} A\hat{Q}_2$。
Let $\hat{Q}_2 = 1 \oplus Q_2$. Find $\hat{Q}_2^{-1} A\hat{Q}_2$.
令
$$ A = \begin{bmatrix} -1 & -4 & 2 \\ 2 & 4 & -1 \\ 0 & -2 & 3 \end{bmatrix}. $$求一個實垂直矩陣 $Q$ 使得 $Q\trans AQ$ 為一上三角矩陣。
Let
$$ A = \begin{bmatrix} -1 & -4 & 2 \\ 2 & 4 & -1 \\ 0 & -2 & 3 \end{bmatrix}. $$Find a real orthogonal matrix $Q$ such that $Q\trans AQ$ is an upper triangular matrix.
令 $A$ 為一方陣、 $Q$ 為一可逆矩陣、 而 $T$ 為一上三角矩陣。
已知 $Q^{-1}AQ = T$,
證明 $\spec(A) = \spec(D)$ 且它們就是 $D$ 的對角線元素所成的集合。
Let $A$ be a square matrix, $Q$ an invertible matrix, and $T$ an upper triangular matrix.
It is known that $Q^{-1}AQ = T$. Show that $\spec(A) = \spec(D)$ and they equal the set of diagonal entries of $D$.
若 $P$ 和 $Q$ 為大小相同的么正矩陣,證明 $PQ$ 和 $QP$ 都是么正矩陣。
若 $P$ 和 $Q$ 為大小相同的實垂直矩陣,證明 $PQ$ 和 $QP$ 都是實垂直矩陣。
If $P$ and $Q$ are unitary matrices of the same order, show that $PQ$ and $QP$ are both unitary. If $P$ and $Q$ are real orthogonal matrices of the same order, show that $PQ$ and $QP$ are both real orthogonal.
證明一般版本的薛爾上三角化定理:
Let $A$ be a square complex matrix.
Then there are a unitary matrix $Q$ such that $Q^*AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
Prove the Schur triangulation theorem.
Let $A$ be a square complex matrix.
Then there are a unitary matrix $Q$ such that $Q^*AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
證明所有特徵值皆為實數的實矩陣版本的薛爾上三角化定理:
Let $A$ be a square real matrix.
Then there are a real orthogonal matrix $Q$ such that $Q\trans AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
Prove the Schur triangulation theorem for real matrices with real eigenvalues.
Let $A$ be a square real matrix.
Then there are a real orthogonal matrix $Q$ such that $Q\trans AQ = T$ is a upper triangular matrix.
Necessarily, the diagonal entries of $T$ are the eigenvalues of $A$.
證明一般實矩陣版本的薛爾上三角化定理:
Let $A$ be a square real matrix.
Then there are a real invertible matrix $Q$ such that $Q^{-1}AQ = T$ has the form
such that $B_k = \begin{bmatrix} \lambda_k \end{bmatrix}$ if $\lambda_k\in\mathbb{R}$
and $B_k = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ if $\lambda = a + bi$ with $b \neq 0$.
Necessarily, the diagonal blocks of $T$ determine the eigenvalues of $A$.
Prove the Schur triangulation theorem for real matrices.
Let $A$ be a square real matrix.
Then there are a real invertible matrix $Q$ such that $Q^{-1}AQ = T$ has the form
such that $B_k = \begin{bmatrix} \lambda_k \end{bmatrix}$ if $\lambda_k\in\mathbb{R}$
and $B_k = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ if $\lambda = a + bi$ with $b \neq 0$.
Necessarily, the diagonal blocks of $T$ determine the eigenvalues of $A$.