# Is Schur factorization unique?

## Is Schur factorization unique?

Although every square matrix has a Schur decomposition, in general this decomposition is not unique. It is clear that if A is a normal matrix, then U from its Schur decomposition must be a diagonal matrix and the column vectors of Q are the eigenvectors of A.

What special form does Schur’s theorem take when A is unitary?

Schur’s unitary triangularization theorem says that every matrix is unitarily equivalent to a triangular matrix.

### What is Triangularization of matrix?

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

What is hessenberg decomposition?

A Hessenberg decomposition is a matrix decomposition of a matrix into a unitary matrix and a Hessenberg matrix such that. where. denotes the conjugate transpose. Hessenberg decomposition is implemented in the Wolfram Language as HessenbergDecomposition[m].

#### What is Jordan decomposition?

The Jordan matrix decomposition is the decomposition of a square matrix into the form. (1) where and are similar matrices, is a matrix of Jordan canonical form, and is the matrix inverse of . In other words, is a similarity transformation of a matrix. in Jordan canonical form.

What is Schur stable?

Introduction. A Square complex matrix A is said to be Schur stable if the spectrum of A is contained in the open unit disk in the complex plane. This property plays an important role in the stability theory for discrete time dynamical Systems.

## What is the determinant of a unitary matrix?

UH=U−1. The magnitude of determinant of a unitary matrix is 1.

What do you mean by diagonal matrix?

From Wikipedia, the free encyclopedia. In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero.

### Who is hessenberg?

Hessenberg may refer to: People: Gerhard Hessenberg (1874–1925), German mathematician. Karl Hessenberg (1904–1959), German mathematician and engineer. Kurt Hessenberg (1908–1994), German composer and professor at the Hochschule für Musik und Darstellende Kunst in Frankfurt am Main.

Which is the theorem of Schur’s unitary triangularization?

Schur’s unitary triangularization theorem says that every matrix is unitarily equivalent to atriangular matrix. Precisely, it reads as follows. Theorem 4. GivenA2 Mnwith eigenvalues; : : : ; , counting multiplicities, there existsa unitary matrixU2 Mnsuch that 2x x3

#### Why is the Schur decomposition called a triangulation?

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.

Which is the simplest theorem for triangularization of a matrix?

The simplest such theorem is the Schur triangularization theorem. This says that every matrix is unitarily similar to an upper triangular matrix. Our aim here is to show that though it is very easy to prove it, this theorem has many interesting conse­ quences. Proof of Schur’s Theorem We want to show that given an n x n matrix A, there ex­

## How is the Schur decomposition related to linear algebra?

Schur decomposition. In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows to write an arbitrary matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.