The Schur decomposition of a complex matrix $A$ is of the form $A = Q U Q^*$, where matrix $Q$ is unitary and $U$ is an upper triangular matrix whose diagonal elements are the eigenvalues of $A$.
Questions tagged [schur-decomposition]
58 questions
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Convergence of the complex QR algorithm to Schur decomposition
I study the complex Schur decomposition of a complex matrix $A \in \mathbb{C}^{n \times n}$, that is:
$$ A = U T U^H $$
where $T$ is upper-triangular (the eigenvalues of $A$ appear on its diagonal, and may be complex) and $U$ is unitary $U^H =…
Triceratops
- 404
4
votes
3 answers
Real Schur decomposition of orthogonal matrix
The real Schur decomposition theorem states that for any matrix $A\in\mathbb R^{n\times n}$, there exists an orthogonal matrix $Q$ and a "quasitriangular" matrix $T$ such that $A=QTQ^T$. Here, "quasitriangular" means that $T$ has the…
ViHdzP
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Matrix exponential using the Schur decomposition
I have a Hermitian $m\times m$ matrix, say $A$. I can use Schur decomposition and transform the matrix in to $A=QTQ^{\dagger}$. Is it then possible to calculate straightforward the matrix exponential using $\exp[A]=Q\cdot\exp[-a T]\cdot…
user123
- 57
4
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1 answer
Schur decomposition for $3 \times 3$ matrix
Suppose $A=\begin{bmatrix} 1 &-2 &2\\-1 &1 &1\\-2 &0 &3 \end{bmatrix}$, what is the Schur decomposition?
The eigenvalues of $A$ are $\lambda_1 = 1,\lambda_2 = 2+2i$ and $\lambda_3 = 2-2i$. For $2 \times 2$ matrix, if the eigenvalues are real, then I…
Keenan
- 69
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Bound the Condition Number
Let $\mathbf{A} \in \mathbb{C}^{n \times n}$ be a non-zero matrix with Schur decomposition $\mathbf{A}=\mathbf{U}(\boldsymbol{\Lambda}+\mathbf{N}) \mathbf{U}^*$ where $\mathbf{U}$ is unitary, $\mathbf{N}$ is strictly upper triangular, and…
Kai Yue
- 109
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1 answer
Schur decomposition
If $A$ is real and nonsymmetric with Schur decomposition $UTU^H$, then what types of matrices are $U$ and $T$? How are the eigenvalues of $A$ related to $U$ and $T$?
piere
- 95
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QR factorization and Schur decomposition
The $QR$ factorization provides us with a way to write every real matrix $A$ in the form of $QR$, with $Q$ being an orthogonal matrix and $R$ being an upper triangular matrix. I believe that there should also exist a complex version of the $QR$…
Sam Wong
- 2,481
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0 answers
Prove using Schur decomposition $\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1$
a) Prove using Schur decomposition $\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1$ where A is in $\mathbb{C}^{mxm}$ and $p$ is the spectral radius.
b) $\lim_{t \rightarrow \infty}||e^{At}||=0 \iff a(A)<1$ where A is in $\mathbb{C}^{mxm}$ and $a$…
MathIsHard
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Conditions for Schur decomposition and its generalization
Let $M$ be a $n$ by $n$ matrix over a field $F$.
When $F$ is $\mathbb{C}$, $M$ always has a Schur decomposition, i.e. it is always unitarily similar to a triangular matrix, i.e. $M = U T U^H$ where $U$ is some unitary matrix and $T$ is a triangular…
Tim
- 49,162
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1 answer
Schur decomposition to show matrix has $n$ orthonormal eigenvectors
From Gilbert Strang's "Introduction to Linear Algebra." We are trying to show by Schur decomposition that all symmetric matrices are diagonalizable. We write down the Schur decomposition as $A=QTQ^{-1}$ where $A$ is square, $T$ is upper triangular…
Zelazny
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Eigenvalues of a squared symmetric matrix
In Page 185 here it says
... $M^2 y=\sigma^2y$. Since $M$ is symmetric, it follows that $y$ is an eigenvector of $M$ with eigenvalue $\pm \sigma$.
It seems to contradict the example here.
What am I missing?
Dan Feldman
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Is the QZ decomposition (generalized schur decomposition) a continuous mapping?
I am not familiar with this topic and have a naive question about QZ decomposition, which is defined as
For any matrix A and B in $\mathbb{R}^{n\times n}$, there exists orthogonal Q and Z, s.t. $QAZ=T$ upper-quasi triangular and $QBZ=S$ upper…
Kryvtsov
- 151
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votes
1 answer
Motivation behind the key step in proof of the Schur decomposition
I often find myself forgetting how to prove that every square matrix having a Schur factorization because I never really understood the motivation behind the steps, I only memorized how to do it. I would like to rectify this. Here's the proof I am…
Helix
- 307
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1 answer
Schur decomposition nonnegative real numbers on the diagonal
Is it possible to have a Schur decomposition of a matrix $A=URU^H$ so that the upper triangular matrix $R$ only has real non-negative numbers on the diagonal?
I realize the diagonal of $R$ is typically made up of the eigenvalues of $A$ which are not…
Josh
- 175
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Optimisation of nonlinear matrix objective with semidefinite constraints (classical Fisher information)
Given the invertible symmetric block diagonal matrices $S = S_1\oplus S_2$ and $dS = dS_1 \oplus dS_2$, I want to
find the symmetric matrix X with blocks
$$
X = \left[
\begin{array}[cc]
~X_1 & X_{12}\\
X_{12}^T & X_2
\end{array}
\right]
$$
that…
Moha
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