Dan Diebolt has provided this about Matrix Pencils:

Let A and B be two n by n matricies. The set of all matricies of the form A-lambda*B with lambda complex is said to be a matrix pencil. The eigenvalues of the pencil are elements of the set lambda(A,B) defined by

lambda(A<B) = {z complex | det(A-z*B) = 0}

This is taken from Matrix Computations by Golub and Van Loan.

As you can see this is the set up for the generalized eigenvalue problem


I think the term was coined by Kronecker or Grantmacker. Kronecker invented about the underlying theory and Grantmacker wrote about it. I don't know who coined the term off hand but I can find out.

Here are some references that explicitly use the term matrix pencil in their title:

"Roots of Matrix Pencils A*y=lambda*B*y By Existence, Calculations and Relations to Game Theory" Lin. Alg. & Its Applic. 5 207-26

A Note on the Efficient Solution of Matirx Pencil Systems BIT 18, 276-81

If you are wondering why I am such an authority (I am not really) is that I work for a company that makes a circuit simulator: Saber. Linearized circuit equations take the form:

(C*s+G) * v = i

where C is the capacitance matrix, G is the conductance matrix and v and i are the voltage and current vectors. Unless every node has a capacitace tied to it the above equations will have a singular capacitance matrix and the eigenvalues problem becomes a generalized eigenvalue problem. Hence the connection to matrix pencils. Small world 'huh?

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