# 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 M = \begin{pmatrix}0 & \lambda_1 & 0 & 0 & 0 &\cdots\\ -\lambda_1 & 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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### 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 \sum_{i,j=1}^{n}|a_{ij}|^2=\sum_{i=1}^{n}|\lambda_i|^2\implies A\text{ ...
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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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