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Questions tagged [schur-decomposition]

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 $A$.

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Show that $U^* A U$ is upper triangular, with $A$ upper triangular, and $U$ unitary and lower-Hessenberg

Let $A \in \mathbb{C}^{n \times n}$ be upper triangular. Let $u$ be a unit-norm eigenvector of $A$ whose eigenvalue is $a_{11}$ (the top-left entry of $A$). Let $U \in Unitary(n)$ be a lower-...
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Sort Skew-Symmetric Tridiagonal Matrix

Suppose I have a skew-symmetric tridiagonal matrix of the from \begin{equation} M = \begin{pmatrix}0 & \lambda_1 & 0 & 0 & 0 &\cdots\\ -\lambda_1 & 0 &\...
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circulant Schur decomposition

Let $A_{1},A_{2},\dots,A_{m}$ be arbitrary $n\times n$ complex matrices. Prove that there are $n\times n$ unitary matrices $Q_{1},Q_{2},\dots,Q_{m}$ such that matrices $$Q_{1}^{*}A_{1}Q_{2},\ Q_{2}^{*...
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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 ...
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Complex Schur decomposition

The complex Schur decomposition goes as follows: For all $A \in \mathbb{C}^{n \times n}$ there exists a unitary matrix $U$ such that $U^{*}AU$ is triangular say, $U^{*}AU=T$. I have seen it being ...
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Proof of existence Schur decomposition

I was reading proof of existence of Schur decomposition. I understand everything except one thing. Why submatrices like $A_2$ have same eigen value with main matrix $A$ like $\lambda_2$ ? A ...
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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{...
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Equality in the Frobenius norm related to the complex Schur decomposition

Let $A \in \mathbb{F}^{n \times n}$, let $X \in \mathbb{F}^{n \times n}$ and let $X=UTU^{*}$ be the complex Schur decomposition, then does the following equality always hold $$ \| A - UTU^{*} \|^{2}_{...
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How can I express a linear matrix inequality in an expanded form?

In the paper Kalman filtering with intermittent observations by Sinopoli et al., I found the following linear matrix inequality (LMI) $$ \begin{bmatrix}X - (1-\lambda)AXA^T & \sqrt{\lambda}F \\ \...
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Every complex square matrix is unitarily similar to a lower triangular matrix?

Is every complex square matrix is unitarily similar to a lower triangular matrix? I know that by Schur's Lemma, upper triangular matrices would suffice, but what about lower. Intuitively, I think it ...
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Gradient of QZ decomposition

Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
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Understanding form of eigenvalues in a schur decomposition.

I am reading the book Matrix Computations from Golub and Van Loan to try understanding a bit about matrix computations. I stumble across the following results : Lemma : If $A ∈ C^{n×n}$, $B ∈ C^{p×p}$,...
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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=...
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Advanced Book on Linear Algebra

After I took a Linear Algebra class I often found many Linear Algebra results that weren't covered in the class. I would like to learn these results therefore I am looking for a book, or even Notes ...
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Complex Schur Factorization

Find a $\mathbb{C}^{n\times n}$ Schur factorizarion for $A$. $$ A=\begin{bmatrix}1 & 0 & 2\\0 & 5 & 0\\2 & 0 &1 \end{bmatrix} $$ I've found the real Schur factorization $QTQ^*...
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Is there are relationship between the eigenvectors and the real Schur vectors of a real skew-symmetric matrix?

A real skew-symmetric matrix $A$ can be diagonalized with complex eigenvectors and pure imaginary eigenvalues: $$A=V S V^*$$ where $S$ is: $$S = \begin{pmatrix} -\lambda_1\mathrm{i} & 0 & 0 &...
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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 $A=QTQ^T$. ...
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Construction of an Orthogonal matrix to reorder diagonal blocks of real Schur form of a matrix

I have been reading "BLOCK ALGORITHMS FOR REORDERING STANDARD AND GENERALIZED SCHUR FORMS" by DANIEL KRESSNER which states that for a real Schur matrix $$A = \begin{bmatrix}A_{11} & A_{...
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Prove the existence of a similar matrix

$A$, $B$ are two matrices such that $A\ge0$ and $B\ge0$ and either $A>0$ or $B>0$. I am trying to show that matrix $BA$ is similar to a matrix with non-negative diagonal elements. Here; $A$ and $...
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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 ...
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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 ...
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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 ...
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What does algebraically closed field play the role in Schur's unitary triangulation theorem?

Schur's unitary triangulation theorem said that Theorem (Schur’s Triangularization Theorem) Every square complex matrix A is unitarily similar to an upper-triangular matrix, i.e., there exists a ...
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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_{...
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Normality result using Schur form

Let $A\in\mathbb{C}^{n\times n}$ have eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$ Using Schur Form so that \begin{equation}\sum_{i,j=1}^{n}|a_{ij}|^2=\sum_{i=1}^{n}|\lambda_i|^2\implies A\text{ ...
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Schur decomposition of matrix $X - \alpha \, u_1 u_1^T$

Consider the Schur decomposition $X = URU^T$ of a real matrix $X$, where $U$ is orthogonal and $R$ is upper triangular. Is there a nice way to compute the Schur decomposition of the matrix $X - \alpha ...
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Example of Schur decomposition

Find the Schur decomposition for the matrix $$ A=\begin{bmatrix}1&1\\ -2&-1\end{bmatrix}$$ I have attempted to do as the question asks via finding the eigenvectors ($u_1=(-1-i,2)^T$ ...
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Please help me understand this linear algebra proof.

Please help me understand the proof below. Here are my questions: 1.) what does r$_1$ represent? 2.) is $A_1$ just matrix $A$ with the first row and first column deleted? 3.) what does $\tilde{r_1}$ ...
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Any example of the Schur decomposition?

I have been searching the internet for 2 hours and literally found no example of the Schur decomposition, which says that in some conditions on matrix $A$ we can write it as: $$A = Q T Q^H$$ where $T$ ...
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$A$ is a symmetric positive-definite matrix it has square root using SVD

What I want: If $A$ is SPD then there is a matrix $X$ such that $A = X^2$ I am able to prove this using the Schur Decomposition, but I was asked to prove it using SVD decomposition. I was trying to ...
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Proving $\sum \frac{a+b}{c} \geq 2.\sqrt{(a+b+c)(\frac{a}{bc} +\frac{b}{ca}+ \frac{c}{ab})}$

Problem. (Le Khanh Sy) For $a,b,c>0.$ Prove$:$ $$\sum \dfrac{a+b}{c} \geq 2\sqrt{(a+b+c)\Big(\dfrac{a}{bc} +\dfrac{b}{ca}+ \dfrac{c}{ab}\Big)}$$ My proof. After squaring ... it's $$4\,{b}^{2}{c}^{2}...
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Schur decomposition of a "flipped" triangular matrix

I would like to take a general 5 x 5 matrix which is a triangular matrix "upside down", i.e. $$A = \begin{pmatrix} 0 & 0 & 0 & 0 & a\\ 0 & 0 & 0 & b & c\\ 0 &...
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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, ...
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Codable Algorithm to the generalised Complex Schur Decomposition without using Inbuilt functions

So i have been looking online for a proper algorithm from scratch which solves the complex generalised Eigen Value problem for low rank non invertible square matrices. I am aware about the generalised ...
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How to prove the changing of basis matrices is unitary matrices

Suppose A is matrix defined on C I need to prove that A is written as A=OTO* T is triangular matrix O* is Conjugate transpose Is it enough (and how) if I proved that O is unitary that mean O* = (O ...
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How does MATLAB compute the real Schur decomposition?

Let matrix A be defined as ...
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2 by 2 Schur Decomposition

Let $A$ be a real matrix \begin{bmatrix} w & x \\ y & z \end{bmatrix} with complex eigenvalues $a+bi$ and $a-bi$. We're looking for an algorithm to find the Givens rotation matrix ...
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Schur factorization from spectral decomposition

Let $A \in \mathbb C^{d \times d}$. The Schur factorization of $A$ is the decomposition $A = U T U^\ast$, where $T \in \mathbb C^{d \times d}$ is an upper triangular matrix and $U \in \mathbb C^{d \...
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Schur decomposition applications [closed]

I know that Schur decomposition is important in matrix theory and linear algebra. I am doing a research and wondering: Why is it that important? What are some applications of it outside the math ...
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Schur decomposition and upper triangular matrix

Can someone please help me with this problem. If $A\in \mathbb C^{n\times n}$ has distinct eigenvalues. How do I show that if $Q^*AQ=T$ is the Shur decomposition and $AB=BA$, then $Q^*BQ$ is upper ...
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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 \...
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Schur Complement and Schur Decomposition

What is the relationship between Schur's decomposition and complement? Did Schur discover them together / are they used in tandem for anything?
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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 ...
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Is this proof on the Schur decomposition wrong?

https://en.wikipedia.org/wiki/Schur_decomposition In this wikipedia article, the proof of the Schur decomposition is as follows: For a given eigenvalue $ \lambda_i$, we obtain orthonormal ...
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Correspondence of Schur Decomposition and Spectral Decomposition needs P.D?

According to Wikipedia, [...] the Schur decomposition extends the spectral decomposition. In particular, if $A$ is positive definite, the Schur decomposition of $A$, its spectral decomposition, ...
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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 $\...
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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}...
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Schur decomposition of a matrix.

Let $E$ be a symmetric matrix then is it possible to find a unitary matrix $U$ such that the diagonal entries of $U^*EU$ are zero?
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Schur Decomposition Upper Triangular Matrix Partition

On page 21 of Matrix Differential Calculus by Magnus and Neudecker (3rd ed, ISBN:0-471-98632-1), the book states, without any apparent justification, that to prove the statement: If $A$ has $r$ non-...
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Preferred matrix decomposition

Consider a complex square matrix $A\in\mathcal{C}^{n\times n}$. Now let us discuss two kinds of the factorization of $A$, say, eigendecomposition and Schur decomposition because both of them are often ...
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