# Questions tagged [generalized-eigenvector]

This tag is for questions relating to Generalized-eigenvector, a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.

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### Perron-Frobenius theorem for reducible non-negative matrices

Let $M$ be a non-negative matrix ($M_{ij}\geq 0$ for all $i,j$). If $M$ is irreducible, then we know that there exists an eigenvalue $\lambda$ of $M$ that equals the spectral radius $\rho(M)$ and has ...
• 1,041
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### Why do GLS and ML estimators coincide for the estimation of a VAR(p) model?

When estimating the coefficients in a VAR(p) model (assuming normality), the coefficient estimators using GLS and MLE coincide. Could anyone explain why this is the case?
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### A basis that makes a matrix triangular.

Find a basis for $\mathbb C^3$ so that the following matrix is in triangular form: \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} What are the eigenvalues? I ...
• 1,161
1 vote
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### Check if adjacency tensor of hypergraph is irreducible

Consider an m-uniform, n-dimensional hypergraph. It's adjacency tensor is an $\overbrace{(n \times n \times ... \times n)}^{m}$ -dimensional tensor $T$. As shown in [1], to calculate the $(H)$-...
• 279
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### How can I guarantee that all the number of generalized eigenvectors are equal to the algebraic multiplicity?

I have found many questions in the Stack but, I couldn't find what I want... Since my major is physics, I'm not good at the terms in mathematics. So it was hard to reach understanding Jordan normal ...
42 views

### Number of distinct eigenvectors in generalized eigenvalue problem $A v = \lambda B v$ (with structure on $A, B$)

Consider the generalized eigenvalue problem $$A v = \lambda B v$$ where $v \in \mathbb{R}^n, A, B \in \mathbb{R}^{n \times n}$. Suppose that $A, B$ are symmetric and positive semi-definite. Suppose ...
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### Generalized characteristic polynomial coefficients

Given two square matrices $A,\,B$ of order $n$, of which therefore all the terms are known, let us define the following polynomial: $$p(x) := \det(A - x\,B)\,.$$ I was wondering if in the literature ...
• 45
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### How does this problem specifically correspond to a generalized eigenvalue problem?

Given two symmetrical square matrices $A,B \in \mathbb R^{n\times n}$ and a rectangular matrix $W\in \mathbb R^{n,k}$. I want to maximize $$\max_{W} \frac{\det (W^TAW)}{\det(W^TBW)}.$$ I read that the ...
• 1,255
1 vote