# 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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### 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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### 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 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 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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### 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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### 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