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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Calculating $\sum_{k=1}^{n}\tau_k + \lfloor\log_{2}n\rfloor$

The divisor matrix $D=(d_{r,s})_{i,j\in\mathbb{N}}$ is defined by $d_{r,s}=1$ if $r$ divides $s$ and 0 otherwise. Raymond Redheffer considered a finite truncations of the divisor matrix. For each ...
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Schur and eigen decomposition

If a matrix $A$ has Schur decomposition $U = S^{-1}AS$ such that $U$ is diagonal, then is the Schur decomposition same as the eigen decomposition, i.e. is $S$ the matrix of eigenvectors of $A$ and are ...
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50 views

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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Suitable stopping criterion for Schur decomposition

I have seen plenty of times in the literature the QR based algorithm to compute the Scur decomposition of a matrix $A$. ...
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35 views

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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123 views

Schur Decomposition example

Find the Schur decomposition for the matrix \begin{equation}A=\begin{bmatrix}1&1\\ -2&-1\end{bmatrix}.\end{equation} I have attempted to do as the question asks via finding the eigenvectors ($...
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51 views

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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Getting the $L_2$ norm of a symmetric matrix using Schur-Jordan decomposition

Prove that for $A \in \mathbb R^{m \times n}$, $\|A\|_2 = σ_{\max}$ using the Schur-Jordan decomposition of the appropriate matrix. My attempt: I have to apply the Schur-Jordan decomposition on $AA^T$...
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Proving that $\|A\|_2 = σ_{\max}$ using the Schur-Jordan decomposition

Prove that for $A \in \mathbb R^{m \times n}$, $\|A\|_2 = σ_{\max}$ using the Schur-Jordan decomposition of the appropriate matrix. I know that $\|A\|_2 = \max_{x\ne 0}\frac{\|Ax\|_2}{\|x\|_2} = \...
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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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162 views

$A$ 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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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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64 views

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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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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537 views

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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487 views

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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689 views

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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909 views

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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743 views

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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Schur Decomposition and $GL_{2}(\mathbb{C})$

Is there an intuitive way to show that any matrix in the general linear group of dimension $2$ of $\mathbb{C}$ has a Schur decomposition? (I'm sure this decomposition is not unique)
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379 views

Impossible Schur Factorizations

I am having trouble finding the schur factorization of the following matrix: $A=\begin{pmatrix}3&8 \\ -2&3 \end{pmatrix}$ I followed an algorithm in the book, as well as computing an answer ...
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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 $T$?
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Schur decomposition of an $n-$by$-n$ matrix

$(\lambda, x)$ is a simple (with multiplicity 1) eigenpair of $A\in \mathbb C_n$ with $x^Hx=1$, $H$ denotes Hermitian. Use Schur decomposition to show that there exists a nonsingular matrix $(x\ \ X)$...
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297 views

A good strategy to find a Schur decomposition for a rank 1 matrix.

Let $u= \begin{pmatrix} 1\\ -1\\ 2 \end{pmatrix}$ and $v= \begin{pmatrix} 2\\ 1\\ -1 \end{pmatrix}$ $E=uv^T= \begin{pmatrix} 2 & 1 & -1\\ -2 & -1 &1 \\ 4 & 2 & -2 \...
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694 views

Find lower triangular matrix using Schur Factorization

I need to find lower triangular matrix using Schur Factorization $A' = U^T A U$ . Actually after factorizing it results upper triangular matrix [using MATLAB] Expecting result could be as such $$A'= \...
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1k views

Diagonalizable unitarily Schur factorization

Let $A$ be $n x n$ matrix. What exactly is the difference between unitarily diagonalizable and diagonalizable matrix $A$? Can that be that it is diagonalizable but not unitarily diagonalizable? What ...
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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 $...