A matrix pencil is an affine one-parameter family of matrices. For square matrices of size , its determinant is a polynomial of degree at most . If is invertible, its leading coefficient is , so at most distinct parameters give a singular matrix. This translates linear dependence in a family of bases into polynomial root counting.
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The term "matrix pencil" refers to a mathematical concept used in the field of linear algebra, particularly in the context of systems of linear equations, control theory, and numerical analysis. A matrix pencil is typically denoted in the form: \[ \mathcal{A}(\lambda) = A - \lambda B \] where: - \(A\) and \(B\) are given matrices, - \(\lambda\) is a complex variable.