# Questions tagged [matrix-pencil]

A matrix pencil in mathematics is a linear equation system, which consists of matrices with complex elements

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### Determining the pencil of two quadrics, but one in general position

Caveat: I am very new to this topic so I am trying to learn. I apologize if this is a very beginner-level question. I am looking at Levin's pencil method as well as Dupont's method to parameterize the ...
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### Residue matrix of $(sE-A)^{-1}$ associated with pole $p_i$?

I found in a paper that the residue matrix of $$f(s)=(sE-A)^{-1}$$ associated with a pole (generalized eigenvalue) $p_i$ is given by $$x_i (y_i Ex_i)^{-1}y_i,$$ where $A,E\in \mathbb{R}^{n\times n}$, ...
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### Properties of blocks in blockmatrix A if the matrix pair (E,A) is regular

I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and ...
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### Canonical form of pairs of matrices over arbitrary field

I'm looking for a canonical form of pairs of matrices over arbitrary field up to equivalence (Calling pairs ($A, B$) and ($A_1, B_1$) over a field F equivalent if invertible C and D exist over F such ...
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### Why there is such a $\mu_0$ such that both $A-\mu_0 B$ and $C+\mu_0 D$ are both nonsingular matrices?

Let $A,B,C,D\in \mathbb{C}^{n\times n}$. If both $A-\lambda B$ and $C-\lambda D$ are both regular matrix pencils, there exists $\mu_0\in \mathbb{C}$ such that both $A-\mu_0 B$ and $C+\mu_0 D$ are ...
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### is this matrix decomposition right

Let $E,A$ be $m\times n$ matrices. Then I need to show there exists $U,V$ orthogonal of suitable order such that UEV=\begin{bmatrix}E_0&E_{k-1}&\times&\times&\times\\ &0&E_{k-...