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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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Definition for finite and infinite zeros of a matrix pencil [closed]

I'm currently studying generalized linear systems of the form $Ey^{\prime}(t) = Ay(t) + Bu(t)$ with controllability pencil pencil $\left[ sE-A \quad B \right]$. In here it is said that $(E,A,B)$ is ...
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Matrix pencil with determinant equal to zero and a symmetry condition

Let $\hat M(\vec x)=\sum_{i=1}^n x_k\hat M_k$ be an $n\times n$ matrix pencil such that $\det M(\vec x)=0$ for every $\vec x$. The $n$ matrices $\hat M_k$ are linearly independent, that is, $\vec M(\...
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Matrix pencil with determinant equal to zero

Let $\sum_{i=1}^n x_k \hat M_k\equiv \hat M(\vec x)$ be a $n\times n$ matrix pencil such that $\det \hat M(\vec x)=0$ for every $\vec x$. Let us also assume that for every vector $\vec y$, there is a ...
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On the equality of the determinant of two matrix polynomials [closed]

I have checked numerically and it seems true that for any three non-zero real matrices $A,B,C$ of $\dim = 4$, and a scalar $z$, it holds that $$\det(z^2 A + z B + C) = \det(z^2 A - z B + C).$$ Is ...
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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-...
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